Advertisements
Advertisements
प्रश्न
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
Advertisements
उत्तर
We have,
9 + 99 + 999 + ... n terms
\[= \left( 9 + 99 + 999 + . . . + \text { to n terms } \right)\]
\[ = \left\{ \left( 10 - 1 \right) + \left( {10}^2 - 1 \right) + \left( {10}^3 - 1 \right) + . . . + \left( {10}^n - 1 \right) \right\}\]
\[ = \left\{ \left( 10 + {10}^2 + {10}^3 + . . . + {10}^n \right) \right\} - \left( 1 + 1 + 1 + 1 . . .\text { n times } \right)\]
\[ = \left\{ 10 \times \frac{\left( {10}^n - 1 \right)}{10 - 1} - n \right\} \]
\[ = \left\{ \frac{10}{9}\left( {10}^n - 1 \right) - n \right\}\]
\[ = \frac{1}{9}\left\{ {10}^{n + 1} - 9n - 10 \right\}\]
APPEARS IN
संबंधित प्रश्न
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
Find the sum of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.
If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.
Find the sum of 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
Three numbers are in A.P. and their sum is 15. If 1, 3, 9 be added to them respectively, they form a G.P. Find the numbers.
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
Find the geometric means of the following pairs of number:
a3b and ab3
If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, \[\frac{y^3 + z^3}{xyz}\] is equal to
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
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:
`2, 4/3, 8/9, 16/27, ...`
Express the following recurring decimal as a rational number:
`0.bar(7)`
Find : `sum_("r" = 1)^oo 4(0.5)^"r"`
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
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 ______.
The third term of G.P. is 4. The product of its first 5 terms is ______.
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 ______.
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
The sum or difference of two G.P.s, is again a G.P.
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 ______.
Let A1, A2, A3, .... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = `1/1296` and A2 + A4 = `7/36`, then the value of A6 + A8 + A10 is equal to ______.
