Advertisements
Advertisements
प्रश्न
Answer the following:
If for a G.P. first term is (27)2 and seventh term is (8)2, find S8
Advertisements
उत्तर
a = (27)2
tn = arn–1
∴ t7 = ar6
∴ ar6 = 82
∴ r6 = `8^2/27^2 = 2^6/3^6`
∴ r = `2/3`
Sn = `("a"(1 - "r"^"n"))/(1 - "r")` for r < 1
∴ S8 = `((27)^2[1 - (2/3)^"s"])/(1 - 2/3)`
= `3.(27)^2 [(3^8 - 2^8)/3^8]`
= `3^7 ((6561 - 256))/3^8`
∴ S8 = `6305/3`
APPEARS IN
संबंधित प्रश्न
Evaluate `sum_(k=1)^11 (2+3^k )`
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.
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.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Which term of the G.P. :
\[\sqrt{3}, 3, 3\sqrt{3}, . . . \text { is } 729 ?\]
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
Find the 4th term from the end of the G.P.
\[\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}\]
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
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.
The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
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.
Let an be the nth term of the G.P. of positive numbers.
Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.
Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
Find the rational numbers having the following decimal expansion:
\[0 . 6\overline8\]
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
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
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
The two geometric means between the numbers 1 and 64 are
For the G.P. if r = − 3 and t6 = 1701, find a.
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 3 years.
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
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, ...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.
Answer the following:
For a sequence , if tn = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common ratio.
Answer the following:
Find `sum_("r" = 1)^"n" (2/3)^"r"`
If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c
The third term of G.P. is 4. The product of its first 5 terms is ______.
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
If `e^((cos^2x + cos^4x + cos^6x + ...∞)log_e2` satisfies the equation t2 – 9t + 8 = 0, then the value of `(2sinx)/(sinx + sqrt(3)cosx)(0 < x ,< π/2)` is ______.
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 ______.
