The first term of an A.P is log^a and second term is log^b .show that the sum of first n terms in (1/2)log[(b^(n(n−1)) /a^(n(−3)) )]


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Question Number 18022 by ibraheem160 last updated on 13/Jul/17

$T h e f i r s t t e r m o f a n A . P i s {l o g}^{a} a n d s e c o n d t e r m i s$ ${l o g}^{b} . s h o w t h a t t h e s u m o f f i r s t n t e r m s i n$ $\frac{1}{2} l o g [\frac{b^{n (n - 1)}}{a^{n (- 3)}}]$
	Answered by prakash jain last updated on 14/Jul/17

	$f i r s t t e r m = a_{1} = \log a$ $d = \log b - \log a$ $S_{n} = \frac{n}{2} (2 a_{1} + (n - 1) d)$ $= \frac{n}{2} (2 \log a + (n - 1) (\log b - \log a))$ $= \frac{1}{2} (2 n \log a + n (n - 1) \log b - n (n - 1) \log a)$ $= \frac{1}{2} (\log a^{2 n} + \log b^{n (n - 1)} - \log a^{n (n - 1)})$ $= \frac{1}{2} (\log b^{n (n - 1)} - [\log a^{n (n - 1)} - \log a^{2 n}]$ $= \frac{1}{2} (\log b^{n (n - 1)} - [\log \frac{a^{n^{2} - n}}{a^{2 n}}])$ $= \frac{1}{2} (\log b^{n (n - 1)} - \log a^{n (n - 3)})$ $= \frac{1}{2} \log \frac{b^{n (n -)}}{a^{n (n - 3)}}$
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