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

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Q.1. Let Sk, k = 1, 2, ….. , 100, denote the sum of the infinite geometric series whose first term is  

Integer Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand the common ratio is  Integer Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE.  Then the value of 

Integer Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis                         (2010)

Ans. (3) 

Sol.   Using Integer Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  we get

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

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

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

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

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

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

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

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

 

 

Q.2. Let a1,a2,a3........, a11 be real numbers satisfying

a1=15, 27–2a2> 0 and ak=2ak–1–ak–2 for k = 3, 4,..........11.

 if   Integer Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE , then the value of

Integer Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis equal to (2010)

Ans. (0)

Sol.   Given that  ak = 2ak –1–ak –2

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

⇒ a, a2 , a3 , ...,a11 are in AP..
If a is the first term and D the commen difference then 
a21 + a22 + ... +a211= 990 (
⇒ 11a 2 + d 2 (12 + 22 + ...+ 102 ) + 2ad (1 + 2 + ...+ 10) = 990

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

⇒ a+ 35d2 + 150d= 90 Using a = 15,
we get 35d2 + 150d +135 = 0 or 7d2 + 30d + 27= 0 
⇒ d +3)(7d +9) = 0 ⇒ d= –3 or – 9/7

then a2 = 15 - 3 = 12 or 15Integer Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ d ≠ – 97

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

 

Q.3. Let a1, a2, a3 .....a100 be an arithmetic progression with  a= 3 and  Integer Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE For any integer n with 1 ≤  n ≤ 20 , let m = 5n. If  Integer Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  does not depend on n, then a2 is              (2011)

Ans. (9)

Sol.  We have  Integer Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEInteger Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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

which will be independent of n if d = 6 or d = 0 For a proper A.P. we take d = 6 then a2 = 3 + 6 = 9

 

Q.4. A pack con tain s n cards n umber ed fr om 1 to n . Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then k – 20 = (JEE Adv. 2013)

Ans.  (5)

Sol.   Let k, k + 1 be removed from pack.

∴ (1 + 2 + 3 + ... + n) – (k + k + 1) = 1224

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

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

for n = 50,  k = 25 ∴ k – 20 = 5

 

Q.5. Let a, b, c be positive integers such that  Integer Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is an integer. If a,b, c are in geometric progression and the arithmetic mean of a, b, c is b + 2, then the value of  Integer Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE   is
 (JEE Adv. 2014)

Ans. (4) 

Sol.   ∵ a, b, c are in G.P

∴ b = ar and c = ar2

Also  Integer Answer Type Questions: Sequences and Series | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  is an integer

⇒ r is an integer

∵  A.M. of a, b, c is b + 2

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

⇒ a + ar + ar2 = 3ar+6

⇒ a (r- 2r + 1)=6

⇒ a (r - 1)2=6 .

∵  a and r are integers
∴ The only possible values of a and r can be 6 and 2 respectively.

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

 

 

Q.6. Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is (JEE Adv. 2015)

Ans. (9) 

Sol.   

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

a7 = a + 6d = 15d

∵  130 < 15d < 140 ⇒ d = 9

(∵ All terms are natural numbers ∴ d ∈N )

 

 

Q.7. The coefficient of x9 in the expansion of (1 + x) (1 + x2) (1 + x3) ... (1 + x100) is (JEE Adv. 2015)

Ans. (8) 

Sol.  In expansion of (1 + x) (1 + x2) (1 + x3) .... (1 + x100) xcan be found in the following ways x9, x1 +2 8, x 2 + 7, x3 + 6, x4 + 5, x1 + 2 + 6, x1 + 3 + 5, x2 + 3 + 4 The coefficient of x9 in each of the above 8 cases is 1. ∴ Required coefficient = 8.

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