Advertisements
Advertisements
Question
Check whether the following sequence is G.P. If so, write tn.
1, –5, 25, –125 …
Solution
Here, t1 = 1, t2 = –5, t3 = 25, t4 = –125, ...
∴ `"t"_2/"t"_1 = (-5)/1` = – 5, `"t"_3/"t"_2 = 25/(-5)` = – 5, `"t"_4/"t"_3 = (-125)/25` = – 5
Since the ratio of any two consecutive terms is a constant, the given sequence is a GP.
Here, a = 1, r = – 5
∴ tn = arn–1 = 1(– 5)n–1
= (– 5)n–1
APPEARS IN
RELATED QUESTIONS
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
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.
Which term of the G.P. :
\[\sqrt{3}, 3, 3\sqrt{3}, . . . \text { is } 729 ?\]
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
Find three numbers in G.P. whose sum is 65 and whose product is 3375.
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
Find the sum of the following series:
9 + 99 + 999 + ... to n 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 \[\frac{3069}{512}\] ?
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 serie to infinity:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Find the sum of the following serie to infinity:
8 + \[4\sqrt{2}\] + 4 + ... ∞
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.
The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
If a, b, c, d are in G.P., prove that:
\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]
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), (b − c), (c − a) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)
If a, b, c are in G.P., then prove that:
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 A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]
If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
For the G.P. if a = `7/243`, r = 3 find t6.
The numbers 3, x, and x + 6 form are in G.P. Find x
The numbers 3, x, and x + 6 form are in G.P. Find nth term
The numbers x − 6, 2x and x2 are in G.P. Find x
For the following G.P.s, find Sn.
p, q, `"q"^2/"p", "q"^3/"p"^2,` ...
For the following G.P.s, find Sn
0.7, 0.07, 0.007, .....
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
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`-3, 1, (-1)/3, 1/9, ...`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
Select the correct answer from the given alternative.
Which term of the geometric progression 1, 2, 4, 8, ... is 2048
Answer the following:
If for a G.P. first term is (27)2 and seventh term is (8)2, find S8
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.
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
