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.
Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015,…
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … `AR^(n-1)` form a G.P, and find the common ratio
If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.
The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.
If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.
The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.
If \[\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}\] (x ≠ 0), then show that a, b, c and d are in G.P.
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Find the sum of the following geometric progression:
1, 3, 9, 27, ... to 8 terms;
Find the sum of the following geometric progression:
4, 2, 1, 1/2 ... to 10 terms.
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:
\[1 - \frac{1}{3} + \frac{1}{3^2} - \frac{1}{3^3} + \frac{1}{3^4} + . . . \infty\]
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
Find the rational number whose decimal expansion is `0.4bar23`.
The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
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, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
If p, q, r, s are in G.P. show that p + q, q + r, r + s are also in G.P.
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after 10 years.
For a G.P. if a = 2, r = 3, Sn = 242 find n
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/2, 1/4, 1/8, 1/16,...`
Express the following recurring decimal as a rational number:
`51.0bar(2)`
Find `sum_("r" = 0)^oo (-8)(-1/2)^"r"`
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
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
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 ______.
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
The sum or difference of two G.P.s, is again a G.P.
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 ______.
If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a0 + a1x + a2x2 + a3x3 ....... then an is ______.
