In a G.P. if the (p+q)th term is m and the (p-q)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, b-a, c-a 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=(b-c)2 then a, b, c are in
If S1, S2, S3, ………Sn 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 S1+S2+S3+ ……Sn is
If a, b, c are in G.P. then the value of (a-b+c)(a+b+c)2-(a+b+c)(a2+b2+c2) is given by
If a1/x=b1/y=c1/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=(b-c)2 then a, b, c are in
If a, b, c are in G.P. then a2+b2, ab+bc, b2+c2 are in
If a, b, c are in G.P then the value of a (b2+c2)-c(a2+b2) is given by
If a, b, c are in G.P. then value of a2b2c2(a-3+b-3+c-3)-(a3+b3+c3) 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 x2, b2, y2 are in
If a, b, c, d are in G.P. then the value of (b-c)2+(c-a)2+(d-b)2-(a-d)2 is given by
If (a-b), (b-c), (c-a) 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 (a-b)2, (b-c)2,(c-a)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, b-a, c-a 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(ab-cd)-(c+a)(b2-c2) is ________
The least value of n for which the sum of n terms of the series 1+3+32+………..is greater than 7000 is _________.
If a, b, c are in A.P. and x, y, z in G.P. then the value of (xb.yc.za) ÷ (xc.ya.zb) is _________.
If a, b, c are the pth, qth and rth terms of a G.P. respectively the value of aq-r. br-p. cp-q is ___________.
If a, b, c are in G.P. then the value of a(b2+c2)-c(a2+b2) 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 Sn 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-(a2+b2+c2)(b2+c2+d2)is ______.
If 1+a+a2+………∞=x and 1+b+b2+……∞=y then 1+ab+a2b2+………∞ = 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+1/32+1/23+1/34+1/25+1/36+……is
If a, b, c are in A.P. and x, y, z in G.P. then the value of xb-c. yc-a. za-b 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(n-1)(2n-1) is divisible by
The value of n2++2n[1+2+3+…+(n-1)] is
The sum of n terms of the series 13/1+(13+23)/2+(12+22+33)/3+……is
3n-2n-1 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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