Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Let there be 2n terms in the given G.P. with the first term being a and the common ratio being r.
According to the question
Sum of all the terms = 5 (Sum of the terms occupying the odd places)
\[\Rightarrow a_1 + a_2 + . . . + a_{2n} = 5 \left( a_1 + a_3 + a_5 + . . . + a_{2n - 1} \right)\]
\[ \Rightarrow a + ar + . . . + a r^{2n - 1} = 5 \left( a + a r^2 + . . . + a r^{2n - 2} \right)\]
\[ \Rightarrow a\left( \frac{1 - r^{2n}}{1 - r} \right) = 5a\left\{ \frac{1 - \left( r^2 \right)^n}{1 - r^2} \right\} \]
\[ \Rightarrow 1 + r = 5 \]
\[ \therefore r = 4\]
APPEARS IN
संबंधित प्रश्न
Evaluate `sum_(k=1)^11 (2+3^k )`
Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … `AR^(n-1)` form a G.P, and find the common ratio
If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.
The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of 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.
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric series:
0.15 + 0.015 + 0.0015 + ... to 8 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 \[39 + 13\sqrt{3}\] ?
The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.
Find the sum of the following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.
Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.
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:
\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]
If a, b, c, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
Find the geometric means of the following pairs of number:
2 and 8
If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.
Write the product of n geometric means between two numbers a and b.
If the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
If for a sequence, tn = `(5^("n"-3))/(2^("n"-3))`, show that the sequence is a G.P. Find its first term and the common ratio
Find four numbers in G.P. such that sum of the middle two numbers is `10/3` and their product is 1
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
For a G.P. if a = 2, r = 3, Sn = 242 find n
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
If S, P, R are the sum, product, and sum of the reciprocals of n terms of a G.P. respectively, then verify that `["S"/"R"]^"n"` = P2
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.
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" = 0)^oo (-8)(-1/2)^"r"`
Select the correct answer from the given alternative.
The common ratio for the G.P. 0.12, 0.24, 0.48, is –
Answer the following:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
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
At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.
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 sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
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 ______.
