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
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
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 `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
Find :
nth term of the G.P.
\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]
The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 terms;
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
The 4th and 7th terms of a G.P. are \[\frac{1}{27} \text { and } \frac{1}{729}\] respectively. Find the sum of n terms of the G.P.
If a, b, c are in G.P., prove that:
(a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
If a, b, c are in G.P., then prove that:
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
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
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
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
For the G.P. if a = `7/243`, r = 3 find t6.
For a G.P. if a = 2, r = 3, Sn = 242 find n
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
The value of a house appreciates 5% per year. How much is the house worth after 6 years if its current worth is ₹ 15 Lac. [Given: (1.05)5 = 1.28, (1.05)6 = 1.34]
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/5, (-2)/5, 4/5, (-8)/5, 16/5, ...`
Find : `sum_("n" = 1)^oo 0.4^"n"`
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the areas of all the squares
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.
Select the correct answer from the given alternative.
Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is –
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
At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
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 ______.
If pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
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 ______.
If the sum of an infinite GP a, ar, ar2, ar3, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, .... is ______.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
