Advertisements
Advertisements
प्रश्न
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
Advertisements
उत्तर
Let a be the first term and r be the common ratio of a G.P.
Given that ap = q
⇒ arp–1 = q ....(i)
And aq = p
⇒ arq–1 = p ....(ii)
Dividing equation (i) by equation (ii) we get,
`(ar^(p - 1))/(ar^(q - 1)) = q/p`
⇒ `(r^(p - 1))/(r^(q - 1)) = q/p`
⇒ `r^(p - q) = q/p`
⇒ r = `(q/p)^(1/(p - q))`
Putting the value of r in equation (i), we get
`a[q/p]^(1/(p- q) xx p - 1)` = q
`a[q/p]^((p - 1)/(p - q))` = q
∴ a = `q * [p/q]^((p - 1)/(p - q))`
Now Tp+q = `ar^(p + q - 1)`
= `q[p/q]^((p - 1)/(p - q)) [q/p]^(1/(p - q)(p + q - 1)`
= `q(p/q)^((p - 1)/(p - q)) * (q/p)^((p + q - 1)/(p - q))`
= `q(p/q)^((p - 1)/(q - q)) * (p/q)^((-(p + q - 1))/(p - q))`
= `q(p/q)^((p - 1)/(p - q) - (p + q - 1)/(p - q))`
= `q(p/q)^((p - 1 - p - q + 1)/(p - q))`
= `q(p/q)^((-q)/(p - q))`
= `q(p/q)^(q/(p - q))`
= `(q^(q/(p - q) + 1))/(p^(q/(p - q))`
= `(q^(p/(p - q)))/(p^(q/(p - q))`
= `[q^p/p^q]^(1/(p - q))`
Hence, the required term = `[q^p/p^q]^(1/(p - q))`.
APPEARS IN
संबंधित प्रश्न
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
Show that the sequence <an>, defined by an = \[\frac{2}{3^n}\], n ϵ N is a G.P.
Find:
the 10th term of the G.P.
\[- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .\]
Which term of the progression 18, −12, 8, ... is \[\frac{512}{729}\] ?
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
Find the sum of the following geometric progression:
2, 6, 18, ... to 7 terms;
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 geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
Find the sum of the following series:
0.6 + 0.66 + 0.666 + .... to n terms
How many terms of the sequence \[\sqrt{3}, 3, 3\sqrt{3},\] ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?
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).
How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?
If S1, S2, ..., Sn are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ..., n respectively, then prove that S1 + S2 + 2S3 + 3S4 + ... (n − 1) Sn = 1n + 2n + 3n + ... + nn.
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
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.
If a, b, c, d are in G.P., prove that:
(a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
If xa = xb/2 zb/2 = zc, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.
If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
Find the geometric means of the following pairs of number:
−8 and −2
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
Check whether the following sequence is G.P. If so, write tn.
`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5))`, ...
Which term of the G.P. 5, 25, 125, 625, … is 510?
The fifth term of a G.P. is x, eighth term of a G.P. is y and eleventh term of a G.P. is z verify whether y2 = xz
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 nth term
For a G.P. if a = 2, r = 3, Sn = 242 find n
For a G.P. If t4 = 16, t9 = 512, find S10
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 tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
Which of the following is not true, where A, G, H are the AM, GM, HM of a and b respectively. (a, b > 0)
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.
Answer the following:
Find the sum of infinite terms of `1 + 4/5 + 7/25 + 10/125 + 13/6225 + ...`
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 ______.
