# Question #6a3be

Mar 22, 2015

$512$

Here are two methods of solution, one algebraic and one arithmetical.

Algebraic

The $k$ th term is $a {r}^{k - 1}$,
so $a {r}^{k - 1} = 162$.

The sum of the first $k$ terms is
${S}_{k} = a \frac{1 - {r}^{k}}{1 - r} = \frac{a - a {r}^{k}}{1 - r} = \frac{a - \left(a {r}^{k - 1}\right) r}{1 - r} = 1562$

Substituting,we get:

$\frac{a - 162 \left(\frac{3}{4}\right)}{1 - \left(\frac{3}{4}\right)} = 1562$

Algebra gives us $a = 512$

Arithmetical

Work backwards from the $k$ th term.

$\frac{1}{r} = \frac{4}{3}$

The $k - 1$ th term is $162 \cdot \frac{4}{3} = 216$.

The sum $216 + 162 < 1562$, so keep working backwards:

The previous term is $216 \cdot \frac{4}{3} = 288$ and the sum is still too small.

Continue until we get: The terms we need are:

$512 + 384 + 288 + 216 + 162$