Advertisements
Advertisements
प्रश्न
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
Advertisements
उत्तर
Let S = 8 + 88 + 888 + ..… up to terms
= 8 [1 + 11 + 111 + ….. up to n terms]
= `8/9[9 + 99 + 999 + .... "up to terms"]`
= `8/9[(10 - 1) + (100 - 1) + (1000 - 1) + ...... "up to n terms"]`
= `8/9[(10 + 100 + 1000 + ....... "up to n terms" - "n")]`
= `8/9[(10(10^"n" - 1))/(10 - 1) - "n"]` ......... `[∵ "s" = ("a" ("r"^"n" - 1))/("r" - 1), "a" = 10, "r" = 10]`
= `8/9 [(10(10^"n" - 1))/9 - "n"]`
= `80/81(10^"n" - 1) - 8/9 "n"`
APPEARS IN
संबंधित प्रश्न
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
Given a G.P. with a = 729 and 7th term 64, determine S7.
Find the sum of the products of the corresponding terms of the sequences `2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2`
Find four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4th by 18.
Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is `1/r^n`.
The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `(3 + 2sqrt2) ":" (3 - 2sqrt2)`.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Find the 4th term from the end of the G.P.
The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.
Find the sum of the following geometric series:
(x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;
Find the sum of the following series:
0.6 + 0.66 + 0.666 + .... to n terms
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that \[S_1^2 + S_2^2\] = S1 (S2 + S3).
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.
Find the sum of the following serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.
If a, b, c are in G.P., prove that:
a (b2 + c2) = c (a2 + b2)
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
If (a − b), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If the 4th, 10th and 16th terms of a G.P. are x, y and z respectively. Prove that x, y, z are in G.P.
If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p − q, q − r, r − s are in G.P.
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
For the G.P. if a = `7/243`, r = 3 find t6.
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
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
For the following G.P.s, find Sn
0.7, 0.07, 0.007, .....
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
If one invests Rs. 10,000 in a bank at a rate of interest 8% per annum, how long does it take to double the money by compound interest? [(1.08)5 = 1.47]
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
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`
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
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
The third term of a G.P. is 4, the product of the first five terms is ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
