Question

The sum of first four terms of an gp is 30 and the sum of the first and the last term is 18 find the terms

Answer 1

Let the first four numbers of GP be
$a \: \: \: ar \: \: \: a {r}^{2} \: \: \: a {r}^{3}$
Given,
$a + ar + a {r}^{2} + {a}^{3} = 30 - - - (1)$
and,
$a + a {r}^{3} = 18 \: - - - (2)$
Dividing (1) and (2),
$\frac{1 + r + {r}^{2} + {r}^{3} }{1 + {r}^{3} } = \frac{30}{18}$
Cross multiplying and Simplifying,
$2 {r}^{3} - 3 {r}^{2} - 3r + 2 = 0 \: - - - (3)$
Guessing for the root to start with we get,
$r = - 1$
$dividing \: (3) \: by \: r + 1 \: we \: get$
$2 {r}^{2} - 5r + 2 = 0$
$this \: gives \: r = 2 \: or \: \frac{1}{2}$
$if \: r = - 1 \: \: a \: is \: not \: defined \: as \: per \: equation \: (2)$
$if \: r = 2 \: \: \: \: \: \: a = 2$
$this \: gives \: gp \: as \: 2 \: \: 4 \: \: 8 \: \: 16$
$if \: r = \frac{1}{2} \: \: \: \: \: a = 16$
$this \: gives \: gp \: as \: 16 \: \: \: 8 \: \: \: 4 \: \: \: 2$