Question

Find the sum to infinity of the given arithmetico-geometric sequences.
$1,\frac{-3}{2}, \frac{5}{4}, \frac{-7}{8}, \frac{9}{16}, \frac{-11}{32},...$

Answer 1

Answer:

Sum of the given AGP is

$\bf\,S_{\infty}=\frac{2}{9}$

Step-by-step explanation:

Given AGP is

$1,\frac{-3}{2}, \frac{5}{4}, \frac{-7}{8}, \frac{9}{16}, \frac{-11}{32},...$"

Here

$a=1$

$d=2$

$r=\frac{-1}{2}$

$\text{The sum of infinite AGP is }$

$\boxed{\bf\,S_{\infty}=\frac{a}{1-r}+\frac{d\,r}{(1-r)^2}}$

$\implies\,S_{\infty}=\frac{1}{1+\frac{1}{2}}+\frac{2(\frac{-1}{2})}{(1+\frac{1}{2})^2}$

$\implies\,S_{\infty}=\frac{1}{\frac{3}{2}}+\frac{(-1)}{(\frac{3}{2})^2}$

$\implies\,S_{\infty}=\frac{2}{3}+\frac{(-1)}{\frac{9}{4}}$

$\implies\,S_{\infty}=\frac{2}{3}-\frac{4}{9}$

$\implies\,S_{\infty}=\frac{6-4}{9}$

$\therefore\boxed{S_{\infty}=\frac{2}{9}}$

Answer 2

The sum to infinity of the given series is  $\frac{2}{9}$.

Step-by-step explanation:

We are given the following arithmetico-geometric sequence below;

$1,\frac{-3}{2} ,\frac{5}{4} ,\frac{-7}{8} ,\frac{9}{16} ,\frac{-11}{32} ,........$

As we know that the formula for Sum to infinity of an arithmetico-geometric series is given by;

                $S_\infty = \frac{a}{1-r} +\frac{dr}{(1-r)^{2} }$  , where $\left | r \right | < 1$

Here, a = first term of the series = 1

          r = common ratio = $\frac{-1}{2}$   {dividing 1st term and 2nd term}

          d = constant increment in the numerator = 2

This series is a mixture of arithmetic and geometric series.

Now, putting the values in the formula we get;

                  $S_\infty = \frac{1}{1-(\frac{-1}{2}) } +\frac{2 \times (\frac{-1}{2}) }{(1-(\frac{-1}{2}))^{2} }$

                  $S_\infty = \frac{1}{1+\frac{1}{2} } +\frac{(-1) }{(1+\frac{1}{2})^{2} }$

                  $S_\infty = \frac{2}{3 } +\frac{(-1) }{(1+\frac{1}{4}+1) }$

                  $S_\infty = \frac{2}{3 } -\frac{1 }{\frac{9}{4} }$

                  $S_\infty = \frac{2}{3 } -\frac{4 }{9 }$

                  $S_\infty = \frac{6-4}{9 }$  =  $\frac{2}{9}$.

Hence, the sum to infinity of the given series is  $\frac{2}{9}$.