If the sum of the second, third and fourth terms of a positive-term G.P.

Question

If the sum of the second, third and fourth terms of a positive-term G.P. is $3$, and the sum of its sixth, seventh and eighth terms is $243$, then the sum of the first $50$ terms of this G.P. is:

Accepted Answer

Let first term = a > 0

Common ratio = r > 0

ar + ar2
 + ar3
 = 3 ....(i)

ar5
 + ar6
 + ar7
 = 243 ....(ii)

$$ \Rightarrow $$ r4(ar + ar2 + ar3) = 243

$$ \Rightarrow $$ r4(3) = 243

$$ \Rightarrow $$ r = 3 as r > 0

from (i)

3a + 9a + 27a = 3

$$ \Rightarrow $$ a = $${1 \over {13}}$$

$$ \therefore $$ S50 = $${{a\left( {{r^{50}} - 1} \right)} \over {\left( {r - 1} \right)}}$$

= $${1 \over {26}}\left( {{3^{50}} - 1} \right)$$

Solutions · 1