In a G.P. if the (p+q)^{th} term is m and the (pq)^{th} term is n then the pth term is _________.
The sum of n terms of the series 0.5+0.05+0.555+………… is
If a, ba, ca are in G.P. and a=b/3=c/5 then a, b, c are in
The sum of term of the series 5+55+555+…..is
If a, b, (c+1) are in G.P. and a=(bc)^{2} then a, b, c are in
If S_{1}, S_{2}, S_{3}, ………S_{n} are the sums of infinite G.P.s whose first terms are 1, 2, 3……n and whose common ratios are 1/2, 1/3, ……1/(n+1) then the value of S_{1}+S_{2}+S_{3}+ ……S_{n} is
If a, b, c are in G.P. then the value of (ab+c)(a+b+c)^{2}(a+b+c)(a^{2}+b^{2}+c^{2}) is given by
If a^{1/x}=b^{1/y}=c^{1/z} and a, b, c are in G.P. then x, y, z are in
If a, b, (c+1) are in G.P. and a=(bc)^{2} then a, b, c are in
If a, b, c are in G.P. then a^{2}+b^{2}, ab+bc, b^{2}+c^{2} are in
If a, b, c are in G.P then the value of a (b^{2}+c^{2})c(a^{2}+b^{2}) is given by
If a, b, c are in G.P. then value of a^{2}b^{2}c^{2}(a^{3}+b^{3}+c^{3})(a^{3}+b^{3}+c^{3}) is given by
If a, b, c, d are in G.P. then a+b, b+c, c+d are in
If a, b, c are in A.P. a, x, b are in G.P. and b, y, c are in G.P. then x^{2}, b^{2}, y^{2} are in
If a, b, c, d are in G.P. then the value of (bc)^{2}+(ca)^{2}+(db)^{2}(ad)^{2} is given by
If (ab), (bc), (ca) are in G.P. then the value of (a+b+c)^{2}3(ab+bc+ca) is given by
If a, b, c, d are in G.P. then (ab)^{2}, (bc)^{2},(ca)^{2} are in
If a, b, x, y, z are positive numbers such that a, x, b are in A.P. and a, y, b are in G.P. and z=(2ab)/(a+b)then
If a, ba, ca are in G.P. and a=b/3=c/5 then a, b, c are in
If a, b, c, d are in G.P. then the value of b(abcd)(c+a)(b^{2}c^{2}) is ________
The least value of n for which the sum of n terms of the series 1+3+3^{2}+………..is greater than 7000 is _________.
If a, b, c are in A.P. and x, y, z in G.P. then the value of (x^{b}.y^{c}.z^{a}) ÷ (x^{c}.y^{a}.z^{b}) is _________.
If a, b, c are the p^{th}, q^{th} and r^{th} terms of a G.P. respectively the value of a^{qr}. b^{rp}. c^{pq} is ___________.
If a, b, c are in G.P. then the value of a(b^{2}+c^{2})c(a^{2}+b^{2}) is __________
If S be the sum, P the product and R the sum of the reciprocals of n terms in a G.P. then P is the ________ of S^{n} and R^{n}.
If the sum of three numbers in G.P. is 21 and the sum of their squares is 189 the numbers are __________.
The sum of n terms of the series 7+77+777+……is
If a, b, c, d are in G.P. then the value of (ab+bc+cd)^{2}(a^{2}+b^{2}+c^{2})(b^{2}+c^{2}+d^{2})is ______.
If 1+a+a^{2}+………∞=x and 1+b+b^{2}+……∞=y then 1+ab+a^{2}b^{2}+………∞ = x is given by ________.
ANSWER : a
Solution : Given, x=1+a+a^2+......∞
Since this is a infinite G.P. series, where, (first term)=1 and (common difference)=a,
So, x = 1/(1−a)
⇒ x−ax=1
⇒ ax=x−1
⇒ a=(x−1)/x
Similarly, y=1+b+b^2 +......∞ is a infinite G.P. series, where, (first term)=1 and
(common difference)=b,
So, y = 1/(1−b)
⇒ y−by=1
⇒ by=y−1
⇒ b=(y−1)/y
And now,
L.H.S.=1 + ab + a^2b^2 + ....∞
= 1/(1−ab) (infinte G.P. series where (first term)=1 and (common difference)=ab
= 1/{1−(x−1/x)(y−1/y)}
= xy/(xy−xy+x+y−1)
= (xy)/(x+y−1)
Sum upto ∞ of the series 1/2+1/3^{2}+1/2^{3}+1/3^{4}+1/2^{5}+1/3^{6}+……is
If a, b, c are in A.P. and x, y, z in G.P. then the value of x^{bc}. y^{ca}. z^{ab} is ______.
If the sum of three numbers in G.P. is 35 and their product is 1000 the numbers are _________.
Three numbers whose sum is 15 are A.P. but if they are added by 1, 4, 19 respectively they are in G.P. The numbers are _________.
Let the given numbers in A.P. be a – d, a, a + d.
According to question,
Hence, the numbers are 5 – d, 5, 5 + d.
Adding 1, 4 and 19 in first, second and third number respectively, we get
Since these numbers are in G.P.
Hence the numbers are
26, 5, –16 or 2, 5, 8.
n(n1)(2n1) is divisible by
For n=1
n(n+1)(2n+1) = 6, divisible by 6.
Let the result be true for n=k
Then, k(k+1)(2k+1) is divisible by 6.
So k(k+1)(2k+1) =6m (1)
Now to prove that the result is true for n=k+1
That is to prove, (K+1)(k+2)(2k+3) is divisible by 6.
(K+1)(k+2)(2k+3)=(k+1)k(2k+3)+(k+1)2(2k+3)=(k+1)k(2k+1)+(k+1)k2+(k+1)2(2k+3)
=6m+2(k+1)(k+2k+3) using (1)
=6m+2(k+1)(3k+3)
=6m +6(k+1)(k+1)=6[m+(k+1)^2]
So divisible by6.
The value of n^{2}++2n[1+2+3+…+(n1)] is
The sum of n terms of the series 1^{3}/1+(1^{3}+2^{3})/2+(1^{2}+2^{2}+3^{3})/3+……is
3^{n}2n1 is divisible by
The sum of n terms of the series 3+6+11+20+37+……… is
The nth terms of the series is 1/(4.7)+1/(7.10)+1/(10.13)+………is
If the third term of a G.P. is the square of the first and the fifth term is 64 the series would be __________.
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