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Q.1. The harmonic mean of two numbers is 4. Their arithmetic mean A and the geometric mean G satisfy the relation. 2A + G= 27
 Find the two numbers. (1979) 

Ans. 3 and 6 or 6 and 3 

Sol. Let the two numbers be a and b, then

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Also 2A + G2 = 27 ⇒ a + b + ab = 27 ....(2)

Putting ab = 27 – (a+b) in eqn. (1), we get

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEa + b = 9 then ab = 27 –  9 = 18

Solving the two we get a = 6, b = 3 or a = 3, b = 6, which are the required numbers.

 

Q.2. Th e in ter ior an gles of a polygon ar e in ar ith metic progression. The smallest angle is 120°, and the common difference is 5°, Find the number of sides of the polygon. (1980)

Ans. 9

 Sol. Let there be n sides in the polygon.
Then by geometry, sum of all n interior angles of polygon = (n – 2) × 180° Also the angles are in A.P. with the smallest angle = 120° , common difference = 5° ∴ Sum of all interior angles of polygon

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Thus we should have

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ 5n2 + 235n= 360n- 720
⇒ 5n2 - 125n + 720= 0
⇒ n- 25n + 144= 0
⇒ (n -16) (n - 9)= 0
⇒ n = 16,  9

Also if n = 16 then 16th angle = 120 + 15 × 5 = 195° > 180°

∴ not possible. Hence n = 9.

 

Q.3. Does there exist a geometric progression containing 27, 8 and 12 as three of its terms  ? If it exits, how many such progressions are possible ? (1982 - 3 Marks)

Ans. yes, infinite

Sol. If possible let for a G.P.
Tp= 27 = ARp–1 ....(1)
Tq= 8 = ARq–1 ....(2)
Tr= 12 = ARr–1 ....(3)
From (1) and (2)

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE....(4)

From (2) and  (3):

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE....(5)

From (4) and (5):

R = 3/2; p – q = 3 ;   q – r = – 1 p – 2q + r = 4;     p, q, r ∈ N ....(6)

As there can be infinite natural numbers for p, q and r to satisfy equation (6)

∴ There can be infinite G.P’s.

 

Q.4. Find three numbers a, b, c, between 2 and 18 such that (i) their sum is 25 (ii) the numbers 2, a, b sare consecutive terms of an A.P. and (iii) the numbers b, c, 18 are consecutive terms of a G.P. (1983 - 2 Marks)

Ans.  a = 5,  b = 8,  c = 12

Sol. 2 < a, b, c  <  18         a + b + c = 25 ....(1)

2, a, b are in AP ⇒ 2a = b + 2 ⇒ 2a – b = 2 ....(2)

b, c, 18 are in GP ⇒ c2 = 18b ....(3)

From  (2)   Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE+  Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

(3) ⇒ c2 = 6 (48 - 2c) ⇒ c+ 12c - 288= 0

⇒ c = 12, – 24 (rejected)   ⇒ a = 5,  b = 8,  c = 12

 

 

Q. 5. If a > 0, b > 0 and c > 0, prove that

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE         (1984 - 2 Marks)

Ans. Sol. Given that a, b, c > 0

We know for +ve numbers A.M. ≥ G.M.

∴ For +ve numbers a, b, c we get

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE             ....(1)

Also for +ve numbers  Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwe get

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE....(2)

Multiplying in eqs (1) and (2) we get

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEProved.

 

Q.6. If n is a natural number such that

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEan d p1, p2, ....., pk are distinct primes, then show that ln n ≥ k ln2 (1984 - 2 Marks)  

Ans. Sol. Given that  Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE        ....(1)

Where n∈N and  p1,  p2, p3, ......pare distinct prime numbers.
Taking log on both sides of eq. (1),

we get log n = α1 log p1+ α2 log p2 + ....+ ak  log pk ....(2)

Since every prime number is such that

p≥ 2

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE....(3)

∀ i= 1 (1) k Using (2) and (3)

we get log n ≥ α1 log 2 + αlog 2 + α3 log 2 + ....+ αk log 2
⇒ log n ≥ (α12+ ....+ αk) log 2
⇒ log n ≥ k log 2 Proved.

 

Q.7. Find the sum of the series :

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE ... up to  terms]

Ans. Sol. The given series is

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Now,  Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Similarly,   Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Hence the given series is,

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE.... to terms

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE[ Summing the G.P.]

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

Q.8. Solve for x the following equation : (1987 - 3 Marks)  
 log(2 x +3) (6 x2 + 23x + 21) = 4 - log (3x+7) (4 x2 + 12x+ 9)

Ans.  -1/4

Sol. The given equation is log (2x + 3) (6x2 + 23 + 21)
= 4 – log3x+7 (4x2 + 12x + 9)

⇒ log(2x+3) (6x2 + 23x + 21) + log(3x+7) (4x2 + 12x + 9) = 4

⇒ log(2x+3) (2x + 3) (3x + 7) + log(3x+7) (2x + 3)2  x =4

⇒ 1+log(2x+3) (3x + 7) + 2log(3x+7) (2x + 3) =4

⇒ log(2x+3)(3x + 7) +Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Let log(2x+3)(3x + 7) = y ....(1)

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEy2 -3y+2=0

⇒ (y – 1)  (y – 2) = 0 ⇒ y = 1, 2

Substituting the values of y in (1), we get

⇒ log(2x+3)(3x + 7) = 1  and     log(2x+3)(3x + 7) = 2

⇒ 3x + 7 = 2x + 3 an d3x + 7 = (2x + 3)

⇒ x  = – 4 an d4x2 + 9x + 2 = 0

⇒ x = – 4 an d(x + 2) (4x + 1) = 0

⇒ x = – 4 and x = – 2, x = – Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

As log a x is defined for x > 0 and a > 0 (a ≠ 1), the possible value of x should satisfy all of the following inequalities :

⇒ 2x + 3 > 0 and 3x + 7 > 0

Also 2x + 3 ≠ 1 and 3x + 7 ≠ 1

Out of x  =  – 4, x = – 2 an d x  = –Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE only x  = – Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

satisfies the above inequalities.

So only solution is x = –Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

Q. 9. If log 3 2 , log3 (2 x , 5) , and Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEare in arithmetic progression, determine the value of x. (1990 -  4 Marks)

Ans.  3

Sol. Given that log2, log3 (2– 5), log3 (2x –7/2) are in A.P.
⇒ 2 log3 (2x – 5) = log3 + log3 (2x – 7/2)

⇒ (2 x - 5) 2 Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ (2x)2 – 10.2+ 25 – 2.2x + 7 = 0 ⇒ (2x)2 – 12.2+ 32 = 0

Let 2x  = y, then we get, y2 – 12y + 32 = 0
⇒ (y – 4) (y – 8) = 0
⇒ y = 4  or  8
⇒ 2x = 22  or 23 ⇒ x = 2  or 3

But for log(2x – 5) and log3 (2– 7/2) to be defined
2x –  5 > 0 and 2x  –  7/2 > 0 ⇒ 2x  >  5 and 2 > 7/2
⇒ 2> 5 ⇒ x ≠ 2  and therefore x = 3.

 

Q.10. Let p be the first of the n arithmetic means between two numbers and q the first of n harmonic means between the same numbers. Show that q does not lie between p and

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE(1991 -  4 Marks)

Ans. Sol. Let a and b be two numbers and A1, A2, A3, .... An be n A.M’s between a and b.
Then a, A1, A2, ..... An, b are in A.P.
There are (n + 2) terms in the series, so that

a + (n + 1) d = b Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE....(1)

The first H.M. between a and b, when nHM’s are inserted between a and b can be obtained by  replacing a by Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  and b by  Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE in eq. (1) and then taking its reciptocal.

Therefore, Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE...(2)

We have to prove that q cannot lie between p

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Now, n + 1>n –1 Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE....(3)
Now to prove the given, we have to show that q is less than p.

For this, let,  Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ (provided a and b and hence p and q are +ve) p > q            ....(4)

From 3 and (4), we get,  Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ q can not lie between p and Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE p, if a and b are +ve numbers.

 

Q.11. If S1, S2 ,S3, ............... , Sn are the sums of infinite geometric series whose first terms are 1, 2, 3, ..............., n and whose common ratios are

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE respectively,,then find the values of Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE (1991 -  4 Marks)

Ans. Sol.  We have,

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

....................................................
....................................................

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

NOTE THIS STEP :

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

Q.12. The real numbers x1, x2, x 3 s atis fying the equation x, x2 + βx + γ < 0 are in AP. Find the intervals in which β and γ lie. (1996 - 3 Marks)

Ans. Sol. Since x1,  x2,  x3 are in A.P.
Therefore, let  x= a – d, x= a and x3 = a + d and x1,  x2,  x3 are the roots of  x3 – x2 + βx  + γ =0
We have ∑α = a – d + a + a + d = 1 ....(1)
∑αβ = (a – d) a + a ( a + d) + (a – d) (a + d) = b ....(2)
αβγ = (a – d) a (a + d) = – γ ....(3)

From (1), we get, 3a = 1 ⇒ a = 1/3

From (2), we get, 3a2 – d2 = β

⇒ 3(1/3)2 – d2 = β ⇒ 1/3 – β = d

We know that d> 0 ∀ d ∈ R

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE            ∵ d 2 ≥ 0

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

From (3), a (a2 – d2)  = – γ

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

Q.13. Let a, b, c, d be real numbers in G.P. If u, v, w, satisfy the system of equations (1999 - 10 Marks)
   Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 then show that the roots of the equation

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 and 20x+ 10 (a - d)2 x - 9 = 0 are reciprocals of each other.

Ans. 

Sol. Solving the system of equations, u + 2v + 3w = 6, 

4u + 5v + 6w = 12 and 6u + 9v = 4

we get u = – 1/3, v = 2/3, w = 5/3

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Let r be the common ratio of the G.P., a, b, c, d. Then b = ar, c = ar2,  d = ar3 .
Then the first equation

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE+ [(b – c)+ (c – a)+ (d – b)2]x + (u + v + w) = 0 becomes

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

The second equation is, 20 x2 + 10 (a – ar3)2  x – 9 = 0

i.e., 20 x2 + 10 a (1 – r3)x – 9 = 0 ....(2)

Since (2) can be obtained by the substitution x → 1/x , equations (1) and (2) have reciprocal roots.

 

Q.14. The fourth power of the common difference of an arithmatic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.           (2000 - 4 Marks)

Ans. 

Sol.  Let a – 3d, a – d, a + d and a + 3d  be any  four consecutive terms of an A.P. with common difference 2d.

∵ Terms of A.P. are integers, 2d is also an integer.
Hence  P = (2d)4 + (a – 3d) (a – d) (a + d) (a + 3d)

=16 d 4 + (a 2 – 9 d 2) (a 2 – d 2) = (a 2 – 5 d 2)2

Now, a 2 – 5 d 2 = a 2 – 9 d 2 + 4 d 2 = (a – 3 d) (a + 3 d) + (2 d)2 =  some integer

Thus, P = square of an integer.

 

Q.15. Let a1, a2, …, an be positive real numbers in geometric progression. For each n, let An, Gn, Hbe respectively, the arithmetic mean, geometric mean, and harmonic mean of a1, a2, …, an. Find an expression for the geometric mean of G1, G2, …, Gn in terms of A1, A2 , …, An, H1, H2, …, Hn. (2001 - 5 Marks)

Ans.

 Sol. Given that a1, a2, ....an are +ve real no’s in G.P.

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

An is A.M. of a1, a2, ....., an

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE.... (1)  (For r ≠ 1)

Gn is G.M. of a1, a2, ...., an

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE          .... (2)   (r ≠ 1)

His H.M. of a1,  a2, ....., an

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE(r ≠ 1) ...(3)
We also observe that

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ AHn =Gn2....(4)

∴ Now, G.M. of   G1, G2, .... Gn is

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  [Using equation (4)

G = (A1A2....AnH1H2....Hn)1/2n   ...(5)

If  r = 1 then An = Gn = Hn = a

Also An H= Gn2

∴ For r = 1 also, equation (5) holds.

Hence we get, G = (A1A2....AnH1H2....Hn)1/2n

 

Q.16. Let a, b be positive real numbers. If a, A1, A2, b are in arithmetic progression, a, G1, G2, b are in geometric progression and a, H1, H2, b are in harmonic progression,

show that  Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE    (2002 - 5 Marks)

 

Ans. 

Sol. Clearly A1 + A= a + b

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

Q.17. If a, b, c are in A.P., a2, b2, c2 are in H.P., then prove that either a = b = c or a, b, Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE form a G.P.. (2003 - 4 Marks)

Ans. 

Sol. Given that a, b, c are in A.P.
⇒ 2b = a + c ....(1)
and a2, b2, care in H.P.

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ ac2 + bc= a2b + a2c     [∵ a – b = b – c]
⇒ ac (c – a) + b (c – a) (c + a) = 0
⇒ (c – a) (ab + bc + ca) = 0
⇒ either c – a = 0 or ab + bc + ca = 0

⇒ either c = a or (a + c) b + ca = 0 and then from (i)
2b2 + ca = 0

Either a = b = c    Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

i.e. a, b, – c/2 are in G.P. Hence Proved.

 

Q.18. If anSubjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand

bn = 1 – an, then find the least natural number n0 such that

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE (2006 - 6M)

Ans. 

Sol.Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ (– 3)n +1 < 22n–1 For n to be even, inequality always holds. For n to be odd, it holds for n ≥ 7.
∴ The least natural no., for which it holds is 6
(∵ it holds for every even natural no.)

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