Advertisements
Advertisements
प्रश्न
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
Advertisements
उत्तर
Let the required numbers be G1 and G2.
∴ 5, G1, G2, 40 are in G.P.
∴ t1 = 5, t2 = G1, t3 = G2, t4 = 40
∴ t1 = a = 5, t4 = 40
tn = arn–1
∴ t4 = 5(r)4–1
∴ 40 = 5r3
∴ r3 = 8 = 23
∴ r = 2
G1 = t2 = ar = 5 (2) = 10
G2 = t3 = ar2 = 5(2)2 = 20
∴ For resulting sequence to be in G.P. we need to insert numbers 10 and 20.
APPEARS IN
संबंधित प्रश्न
Which term of the following sequence:
`2, 2sqrt2, 4,.... is 128`
If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P.
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.
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
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.
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
If a, b, c are in G.P., prove that:
\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]
If a, b, c are in G.P., prove that the following is also in G.P.:
a3, b3, c3
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 are in G.P., then prove that:
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
If logxa, ax/2 and logb x are in G.P., then write the value of x.
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 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
For the G.P. if a = `7/243`, r = 3 find t6.
For the G.P. if a = `2/3`, t6 = 162, find r.
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 20th term.
The numbers x − 6, 2x and x2 are in G.P. Find x
The numbers x − 6, 2x and x2 are in G.P. Find 1st term
For a G.P. sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of r
Find the sum to n terms of the sequence.
0.5, 0.05, 0.005, ...
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:
`2, 4/3, 8/9, 16/27, ...`
Express the following recurring decimal as a rational number:
`0.bar(7)`
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
The sum of 3 terms of a G.P. is `21/4` and their product is 1 then the common ratio is ______.
Answer the following:
For a sequence Sn = 4(7n – 1) verify that the sequence is a G.P.
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 pth, qth and rth terms of a G.P. are x, y, z respectively. Find the value of xq–r .yr–p .zp–q
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 x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is ______.
Let `{a_n}_(n = 0)^∞` be a sequence such that a0 = a1 = 0 and an+2 = 2an+1 – an + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.
