Advertisements
Advertisements
प्रश्न
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
Advertisements
उत्तर
We have,
5 + 55 + 555+ ... n terms
Taking 5 as common:
\[S_n\] = 5[1 + 11 + 111 + ... n terms]
\[= \frac{5}{9}\left( 9 + 99 + 999 + . . . \text { n terms } \right)\]
\[ = \frac{5}{9}\left\{ \left( 10 - 1 \right) + \left( {10}^2 - 1 \right) + \left( {10}^3 - 1 \right) + . . . + \left( {10}^n - 1 \right) \right\}\]
\[ = \frac{5}{9}\left\{ \left( 10 + {10}^2 + {10}^3 + . . . + {10}^n \right) \right\} - \left( 1 + 1 + 1 + 1 + . . .\text { n times } \right)\]
\[ = \frac{5}{9}\left\{ 10 \times \frac{\left( {10}^n - 1 \right)}{10 - 1} - n \right\} \]
\[ = \frac{5}{9} \left\{ \frac{10}{9}\left( {10}^n - 1 \right) - n \right\}\]
\[ = \frac{5}{81}\left\{ {10}^{n + 1} - 9n - 10 \right\}\]
APPEARS IN
संबंधित प्रश्न
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?
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, b, c are in A.P,; b, c, d are in G.P and ` 1/c, 1/d,1/e` are in A.P. prove that a, c, e are in G.P.
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
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 the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
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.
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Find the sum of the following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
If a, b, c are in G.P., prove that \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.P.
The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
If a, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this 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
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 a = `7/243`, r = 3 find t6.
The numbers 3, x, and x + 6 form are in G.P. Find x
For the following G.P.s, find Sn
0.7, 0.07, 0.007, .....
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
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:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
Answer the following:
Find `sum_("r" = 1)^"n" (2/3)^"r"`
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is ______.
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
