Advertisements
Advertisements
प्रश्न
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
Advertisements
उत्तर
\[\text { As, } a_p = q\]
\[ \Rightarrow a r^\left( p - 1 \right) = q . . . . . \left( i \right)\]
\[\text { Also, } a_q = p\]
\[ \Rightarrow a r^\left( q - 1 \right) = p . . . . . \left( ii \right)\]
\[\text { Dividing }\left( i \right)\text { by } \left( ii \right), \text { we get }\]
\[\frac{a r^\left( p - 1 \right)}{a r^\left( q - 1 \right)} = \frac{q}{p}\]
\[ \Rightarrow r^\left( p - 1 - q + 1 \right) = \frac{q}{p}\]
\[ \Rightarrow r^\left( p - q \right) = \frac{q}{p}\]
\[ \Rightarrow r = \left( \frac{q}{p} \right)^\frac{1}{\left( p - q \right)} \]
\[\text { Substituting the value of r in } \left( ii \right), \text { we get }\]
\[a \left[ \left( \frac{q}{p} \right)^\frac{1}{\left( p - q \right)} \right]^\left( q - 1 \right) = p\]
\[ \Rightarrow a\left[ \left( \frac{q}{p} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \right] = p\]
\[ \Rightarrow a = p \times \left( \frac{p}{q} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \]
\[ \Rightarrow a = p \left( \frac{p}{q} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \]
\[\text { Now, } \]
\[ a_\left( p + q \right) = a r^\left( p + q - 1 \right) \]
\[ = p \left( \frac{p}{q} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \times \left[ \left( \frac{q}{p} \right)^\frac{1}{\left( p - q \right)} \right]^\left( p + q - 1 \right) \]
\[ = p \left( \frac{p}{q} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \times \left( \frac{q}{p} \right)^\frac{\left( p + q - 1 \right)}{\left( p - q \right)} \]
\[ = p \left( \frac{q}{p} \right)^\frac{- \left( q - 1 \right)}{\left( p - q \right)} \times \left( \frac{q}{p} \right)^\frac{\left( p + q - 1 \right)}{\left( p - q \right)} \]
\[ = p \times \left( \frac{q}{p} \right)^\frac{- \left( q - 1 \right)}{\left( p - q \right)} + \frac{\left( p + q - 1 \right)}{\left( p - q \right)} \]
\[ = p \times \left( \frac{q}{p} \right)^\frac{- q + 1 + p + q - 1}{\left( p - q \right)} \]
\[ = p \times \left( \frac{q}{p} \right)^\frac{p}{\left( p - q \right)} \]
\[ = \frac{p \times q^\frac{p}{\left( p - q \right)}}{p^\frac{p}{\left( p - q \right)}}\]
\[ = \frac{q^\frac{p}{\left( p - q \right)}}{p^\frac{p}{\left( p - q \right)} - 1}\]
\[ = \frac{q^\frac{p}{\left( p - q \right)}}{p^\frac{p - p + q}{\left( p - q \right)}}\]
\[ = \frac{q^\frac{p}{\left( p - q \right)}}{p^\frac{q}{\left( p - q \right)}}\]
\[ = \frac{q^{p \times \frac{1}{\left( p - q \right)}}}{p^{q \times \frac{1}{\left( p - q \right)}}}\]
\[ = \left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\]
संबंधित प्रश्न
Evaluate `sum_(k=1)^11 (2+3^k )`
if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
If a and b are the roots of are roots of x2 – 3x + p = 0 , and c, d are roots of x2 – 12x + q = 0, where a, b, c, d, form a G.P. Prove that (q + p): (q – p) = 17 : 15.
Find :
the 8th term of the G.P. 0.3, 0.06, 0.012, ...
Which term of the G.P. :
\[2, 2\sqrt{2}, 4, . . .\text { is }128 ?\]
Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
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 of the following geometric series:
\[\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };\]
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Evaluate the following:
\[\sum^n_{k = 1} ( 2^k + 3^{k - 1} )\]
The 4th and 7th terms of a G.P. are \[\frac{1}{27} \text { and } \frac{1}{729}\] respectively. Find the sum of n terms of the G.P.
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.
Let an be the nth term of the G.P. of positive numbers.
Let \[\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,\] such that α ≠ β. Prove that the common ratio of the G.P. is α/β.
Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.
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.
Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.
If a, b, c, d are in G.P., prove that:
\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]
If a, b, c are in G.P., then prove that:
If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
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 a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then
Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, \[\frac{y^3 + z^3}{xyz}\] is equal to
For the G.P. if a = `2/3`, t6 = 162, find r.
For the following G.P.s, find Sn
0.7, 0.07, 0.007, .....
Find the sum to n terms of the sequence.
0.2, 0.02, 0.002, ...
For a sequence, if Sn = 2(3n –1), find the nth term, hence show that the sequence is a G.P.
If Sn, S2n, S3n are the sum of n, 2n, 3n terms of a G.P. respectively, then verify that Sn (S3n – S2n) = (S2n – Sn)2.
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:
9, 8.1, 7.29, ...
Express the following recurring decimal as a rational number:
`2.3bar(5)`
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
Answer the following:
Find three numbers in G.P. such that their sum is 35 and their product is 1000
Answer the following:
Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.
If pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
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 ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
