Question

The sum of first three terms of g. p is 16 and sum of the next five terms is 128 determine the first term common ration

Answer 1

➡HERE IS YOUR ANSWER⬇

Let us consider the GP series as

$a + ar + a {r}^{2} + a {r}^{3} + a {r}^{4} + a {r}^{5} + \: ...$

where a = the first term of the series and r is the common ratio.

Given that :

$a + ar + a {r}^{2} = 16 \\ \\ or \: \: a(1 + r + {r}^{2} ) = 16 \: \: .....(1) \\ \\ and \\ \\ a {r}^{3} + a {r}^{4} + a {r}^{5} = 128 \\ \\ or \: \: a {r}^{3} (1 + r + {r}^{2} ) = 128 \: \: .....(2) \\ \\ now \: dividing \: \: (2) \: \: by \: \: (1) \: \: we \: \: get \\ \\ {r}^{3} = 8 \\ \\ hence \: \: r = 2. \\ \\ from \: \: (1) \: \: putting \: \: r = 2 \: \: we \: \: get \\ \\ a(1 + 2 + {2}^{2} ) = 16 \\ \\ or \: \: 7a = 16 \\ \\ so \: \: a = \frac{16}{7}$

Therefore, the first term of the GP series is the 16/7 and the common ratio is 2.

⬆HOPE THIS HELPS YOU⬅