Advertisements
Advertisements
Question
In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.
Advertisements
Solution
\[\text { Let a be the first term and r be the common ratio } . \]
\[ \therefore a_3 = 24 \text { and } a_6 = 192\]
\[ \Rightarrow a r^2 = 24 \text { and } a r^5 = 192\]
\[ \Rightarrow \frac{a r^5}{a r^2} = \frac{192}{24}\]
\[ \Rightarrow r^3 = 8 \]
\[ \Rightarrow r^3 = 2^3 \]
\[ \Rightarrow r = 2\]
\[\text { Putting } r = 2 \text { in a }r^2 = 24\]
\[a \left( 2 \right)^2 = 24 \]
\[ \Rightarrow a = 6\]
\[\text { Now }, {10}^{th}\text { term }= a_{10} = a r^9 \]
\[\text { Putting a = 6 and r = 2 in } a_{10} = a r^9 \]
\[ \Rightarrow a_{10} = \left( 6 \right) \left( 2 \right)^9 = 3072\]
\[\text { Thus, the } {10}^{th}\text { term of the G . P . is } 3072 .\]
APPEARS IN
RELATED QUESTIONS
Find the sum to indicated number of terms in the geometric progressions 1, – a, a2, – a3, ... n terms (if a ≠ – 1).
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.
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
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`
Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...
Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?
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.
Find three numbers in G.P. whose sum is 38 and their product is 1728.
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Find the sum of the following geometric series:
`sqrt7, sqrt21, 3sqrt7,...` to n terms
Evaluate the following:
\[\sum^{10}_{n = 2} 4^n\]
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make \[\frac{3069}{512}\] ?
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.
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.
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 a, b, c, d are in G.P., prove that:
(b + c) (b + d) = (c + a) (c + d)
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) 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 a, b, c are in A.P., b,c,d are in G.P. and \[\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\] are in A.P., prove that a, c,e are in G.P.
Find the geometric means of the following pairs of number:
2 and 8
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.
If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
The two geometric means between the numbers 1 and 64 are
Mark the correct alternative in the following question:
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 p2R3 : S3 is equal to
A ball is dropped from a height of 80 ft. The ball is such that it rebounds `(3/4)^"th"` of the height it has fallen. How high does the ball rebound on 6th bounce? How high does the ball rebound on nth bounce?
The numbers x − 6, 2x and x2 are in G.P. Find nth term
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 the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
Find : `sum_("r" = 1)^oo (-1/3)^"r"`
The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of the areas of all the squares
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Answer the following:
For a G.P. if t2 = 7, t4 = 1575 find a
Answer the following:
Find k so that k – 1, k, k + 2 are consecutive terms of a G.P.
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
