Advertisements
Advertisements
प्रश्न
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
Advertisements
उत्तर
Here,
\[\text { First term, } a = \sqrt{3}\]
\[\text { Common ratio, } r = \frac{a_2}{a_1} = \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}} = \frac{1}{3}\]
\[ \therefore \text { nth term } = a_n = a r^{(n - 1)} = \sqrt{3} \left( \frac{1}{3} \right)^{n - 1} \]
\[\text { Thus, the nth term of the given GP is } \sqrt{3} \left( \frac{1}{3} \right)^{n - 1} .\]
APPEARS IN
संबंधित प्रश्न
Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`
Given a G.P. with a = 729 and 7th term 64, determine S7.
Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … `AR^(n-1)` form a G.P, and find the common ratio
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
Find the sum of the following geometric progression:
1, −1/2, 1/4, −1/8, ... to 9 terms;
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
If S1, S2, ..., Sn are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ..., n respectively, then prove that S1 + S2 + 2S3 + 3S4 + ... (n − 1) Sn = 1n + 2n + 3n + ... + nn.
Express the recurring decimal 0.125125125 ... as a rational number.
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.
If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively
\[\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}\]
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
If a, b, c are in G.P., prove that:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If a, b, c are in G.P., then prove that:
If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c are in G.P.
If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.
Write the product of n geometric means between two numbers a and b.
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
Mark the correct alternative in the following question:
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then p2R3 : S3 is equal to
For the G.P. if r = `1/3`, a = 9 find t7
If for a sequence, tn = `(5^("n"-3))/(2^("n"-3))`, show that the sequence is a G.P. Find its first term and the common ratio
For a G.P. If t3 = 20 , t6 = 160 , find S7
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is ______.
Answer the following:
If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.
