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
\[\text { As, } a_p = q\]
\[ \Rightarrow a r^\left( p - 1 \right) = q . . . . . \left( i \right)\]
\[\text { Also, } a_q = p\]
\[ \Rightarrow a r^\left( q - 1 \right) = p . . . . . \left( ii \right)\]
\[\text { Dividing }\left( i \right)\text { by } \left( ii \right), \text { we get }\]
\[\frac{a r^\left( p - 1 \right)}{a r^\left( q - 1 \right)} = \frac{q}{p}\]
\[ \Rightarrow r^\left( p - 1 - q + 1 \right) = \frac{q}{p}\]
\[ \Rightarrow r^\left( p - q \right) = \frac{q}{p}\]
\[ \Rightarrow r = \left( \frac{q}{p} \right)^\frac{1}{\left( p - q \right)} \]
\[\text { Substituting the value of r in } \left( ii \right), \text { we get }\]
\[a \left[ \left( \frac{q}{p} \right)^\frac{1}{\left( p - q \right)} \right]^\left( q - 1 \right) = p\]
\[ \Rightarrow a\left[ \left( \frac{q}{p} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \right] = p\]
\[ \Rightarrow a = p \times \left( \frac{p}{q} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \]
\[ \Rightarrow a = p \left( \frac{p}{q} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \]
\[\text { Now, } \]
\[ a_\left( p + q \right) = a r^\left( p + q - 1 \right) \]
\[ = p \left( \frac{p}{q} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \times \left[ \left( \frac{q}{p} \right)^\frac{1}{\left( p - q \right)} \right]^\left( p + q - 1 \right) \]
\[ = p \left( \frac{p}{q} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \times \left( \frac{q}{p} \right)^\frac{\left( p + q - 1 \right)}{\left( p - q \right)} \]
\[ = p \left( \frac{q}{p} \right)^\frac{- \left( q - 1 \right)}{\left( p - q \right)} \times \left( \frac{q}{p} \right)^\frac{\left( p + q - 1 \right)}{\left( p - q \right)} \]
\[ = p \times \left( \frac{q}{p} \right)^\frac{- \left( q - 1 \right)}{\left( p - q \right)} + \frac{\left( p + q - 1 \right)}{\left( p - q \right)} \]
\[ = p \times \left( \frac{q}{p} \right)^\frac{- q + 1 + p + q - 1}{\left( p - q \right)} \]
\[ = p \times \left( \frac{q}{p} \right)^\frac{p}{\left( p - q \right)} \]
\[ = \frac{p \times q^\frac{p}{\left( p - q \right)}}{p^\frac{p}{\left( p - q \right)}}\]
\[ = \frac{q^\frac{p}{\left( p - q \right)}}{p^\frac{p}{\left( p - q \right)} - 1}\]
\[ = \frac{q^\frac{p}{\left( p - q \right)}}{p^\frac{p - p + q}{\left( p - q \right)}}\]
\[ = \frac{q^\frac{p}{\left( p - q \right)}}{p^\frac{q}{\left( p - q \right)}}\]
\[ = \frac{q^{p \times \frac{1}{\left( p - q \right)}}}{p^{q \times \frac{1}{\left( p - q \right)}}}\]
\[ = \left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\]
APPEARS IN
संबंधित प्रश्न
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`
Given a G.P. with a = 729 and 7th term 64, determine S7.
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.
The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
−2/3, −6, −54, ...
Which term of the G.P. :
\[\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?\]
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
Find the sum of the following geometric series:
1, −a, a2, −a3, ....to n terms (a ≠ 1)
Find the sum of the following serie:
5 + 55 + 555 + ... to n terms;
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
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: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.
Find the rational number whose decimal expansion is `0.4bar23`.
Find the rational numbers having the following decimal expansion:
\[0 . \overline3\]
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the 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 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 pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]
The fractional value of 2.357 is
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
Mark the correct alternative in the following question:
Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then p2R3 : S3 is equal to
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 1st 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
Find: `sum_("r" = 1)^10 5 xx 3^"r"`
Select the correct answer from the given alternative.
If common ratio of the G.P is 5, 5th term is 1875, the first term 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 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
If a, b, c, d are in G.P., prove that a2 – b2, b2 – c2, c2 – d2 are also in G.P.
For a, b, c to be in G.P. the value of `(a - b)/(b - c)` is equal to ______.
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.
The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.
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 ______.
