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-

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)}}