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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.
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
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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