Advertisements
Advertisements
प्रश्न
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
Advertisements
उत्तर
We have,
\[ a_1 = 4, a_2 = - 2, a_3 = 1, a_4 = - \frac{1}{2}\]
\[\text { Now }, \frac{a_2}{a_1} = \frac{- 2}{4} = \frac{- 1}{2}, \frac{a_3}{a_2} = \frac{1}{- 2}, \frac{a_4}{a_3} = \frac{- \frac{1}{2}}{1} = \frac{- 1}{2}\]
\[ \therefore \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = \frac{- 1}{2}\]
\[\text { Thus, } a_1 , a_2 , a_3 \text { and } a_4\text { are in G . P . , where a = 4 and }r = \frac{- 1}{2} .\]
APPEARS IN
संबंधित प्रश्न
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
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.
Find the 4th term from the end of the G.P.
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
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).
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
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 the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
If a, b, c, d are in G.P., prove that:
(a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.
If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a − b, d − c 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.
Insert 5 geometric means between 16 and \[\frac{1}{4}\] .
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
Check whether the following sequence is G.P. If so, write tn.
2, 6, 18, 54, …
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
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 x
For a G.P. If t3 = 20 , t6 = 160 , find S7
For a G.P. If t4 = 16, t9 = 512, find S10
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`2, 4/3, 8/9, 16/27, ...`
Express the following recurring decimal as a rational number:
`51.0bar(2)`
If the common ratio of a G.P. is `2/3` and sum to infinity is 12. Find the first term
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
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
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.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is ______.
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:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
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:
If p, q, r, s are in G.P., show that (pn + qn), (qn + rn) , (rn + sn) are also in G.P.
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 ______.
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
