Advertisements
Advertisements
Question
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
Advertisements
Solution
Let a be the first term and r be the common ratio of G.P.
Then S3 = 125 and S6 =125 + 27 = 152
∴ `"S"_6/"S"_3= 152/125`
∴ `([("a"("r"^6 - 1))/("r" - 1)])/([("a"("r"^3 - 1))/("r" - 1)]) = 152/125`
∴ `("r"^6 - 1)/("r"^3 - 1) = 152/125`
∴ `(("r"^3 - 1)("r"^3 + 1))/("r"^3 - 1) = 152/125`
∴ r3 + 1 = `152/125`
∴ r3 = `152/125 - 1 = 27/125 = (3/5)^3`
∴ r = `3/5`
APPEARS IN
RELATED QUESTIONS
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.
If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.
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:
2, 6, 18, ... to 7 terms;
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
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:
x3, x5, x7, ... to n terms
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
The ratio of the sum of the first three terms to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.
Find the sum :
\[\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .\]
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
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:
\[0 . \overline3\]
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
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 the fifth term of a G.P. is 2, then write the product of its 9 terms.
If A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]
Write the product of n geometric means between two numbers a and b.
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
If x is positive, the sum to infinity of the series \[\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is\]
Check whether the following sequence is G.P. If so, write tn.
2, 6, 18, 54, …
For the G.P. if a = `7/243`, r = 3 find t6.
For the G.P. if a = `2/3`, t6 = 162, find r.
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball
Find GM of two positive numbers whose A.M. and H.M. are 75 and 48
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Answer the following:
If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
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 ______.
