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 .
संबंधित प्रश्न
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
For what values of x, the numbers `-2/7, x, -7/2` are in G.P?
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
If a and b are the roots of are roots of x2 – 3x + p = 0 , and c, d are roots of x2 – 12x + q = 0, where a, b, c, d, form a G.P. Prove that (q + p): (q – p) = 17 : 15.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.
If \[\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}\] (x ≠ 0), then show that a, b, c and d are in G.P.
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
The sum of first three terms of a G.P. is \[\frac{39}{10}\] and their product is 1. Find the common ratio and the terms.
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
Find the sum :
\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]
Find the sum of the following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
If a, b, c are in A.P., b,c,d are in G.P. and \[\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\] are in A.P., prove that a, c,e are in G.P.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
For the G.P. if r = − 3 and t6 = 1701, find a.
For the G.P. if a = `2/3`, t6 = 162, find r.
A ball is dropped from a height of 80 ft. The ball is such that it rebounds `(3/4)^"th"` of the height it has fallen. How high does the ball rebound on 6th bounce? How high does the ball rebound on nth bounce?
The numbers 3, x, and x + 6 form are in G.P. Find x
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after n years.
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/2, 1/4, 1/8, 1/16,...`
Express the following recurring decimal as a rational number:
`2.bar(4)`
Express the following recurring decimal as a rational number:
`2.3bar(5)`
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
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 ______.
The sum of infinite number of terms of a decreasing G.P. is 4 and the sum of the terms to m squares of its terms to infinity is `16/3`, then the G.P. is ______.
