Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Given:
\[ S_1 , S_2 , . . . , S_n\text { are the sum of n terms of an G . P . whose first term is 1 in each case and the common ratios are } 1, 2, 3, . . . , n . \]
\[ \therefore S_1 = 1 + 1 + 1 + . . .\text { n terms } = n . . . \left( 1 \right)\]
\[ S_2 = \frac{1\left( 2^n - 1 \right)}{2 - 1} = 2^n - 1 . . . \left( 2 \right)\]
\[ S_3 = \frac{1\left( 3^n - 1 \right)}{3 - 1} = \frac{3^n - 1}{2} . . . \left( 3 \right)\]
\[ S_4 = \frac{1\left( 4^n - 1 \right)}{4 - 1} = \frac{4^n - 1}{3} . . . \left( 4 \right)\]
\[ S_n = \frac{1\left( n^n - 1 \right)}{n - 1} = \frac{n^n - 1}{n - 1} . . . . . . . . . . . . \left( n \right)\]
\[\text { Now, LHS } = S_1 + S_2 + 2 S_3 + 3 S_4 + . . . + \left( n - 1 \right) S_n \]
\[ = n + 2^n - 1 + 3^n - 1 + 4^n - 1 + . . . + n^n - 1 \left[ \text { Using } \left( 1 \right), \left( 2 \right), \left( 3 \right), . . . , \left( n \right) \right]\]
\[ = n + \left( 2^n + 3^n + 4^n + . . . + n^n \right) - \left[ 1 + 1 + 1 + . . . + \left( n - 1 \right) \text { times } \right]\]
\[ = n + \left( 2^n + 3^n + 4^n + . . . + n^n \right) - \left( n - 1 \right)\]
\[ = n + \left( 2^n + 3^n + 4^n + . . . + n^n \right) - n + 1\]
\[ = 1 + 2^n + 3^n + 4^n + . . . + n^n \]
\[ = 1^n + 2^n + 3^n + 4^n + . . . + n^n \]
= RHS
Hence proved .
APPEARS IN
संबंधित प्रश्न
Evaluate `sum_(k=1)^11 (2+3^k )`
Given a G.P. with a = 729 and 7th term 64, determine S7.
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
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`.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
Evaluate the following:
\[\sum^{11}_{n = 1} (2 + 3^n )\]
Find the sum of the following series:
7 + 77 + 777 + ... to n terms;
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
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 log a, log b, log c are in A.P.
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, c, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
If a, b, c are in G.P., prove that the following is also in G.P.:
a3, b3, c3
If a, b, c, d are in G.P., prove that:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
If the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
The two geometric means between the numbers 1 and 64 are
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
Check whether the following sequence is G.P. If so, write tn.
2, 6, 18, 54, …
The fifth term of a G.P. is x, eighth term of a G.P. is y and eleventh term of a G.P. is z verify whether y2 = xz
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
Express the following recurring decimal as a rational number:
`51.0bar(2)`
Answer the following:
If pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of xq–r .yr–p .zp–q
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
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 P2 R3 : S3 is equal to ______.
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 ______.
