Answer
False. For the statement to be true $10-5+\frac{5}{2}-\frac{5}{4}+\text{ }...\text{ }=\frac{10}{1+\frac{1}{2}}$
Work Step by Step
From the given series, we can observe that it is an infinite geometric series and therefore ${{S}_{n}}=\frac{{{a}_{1}}}{\left( 1-r \right)}$.
The provided series is $10-5+\frac{5}{2}-\frac{5}{4}+\text{ }...\text{ }=\frac{10}{1-\frac{1}{2}}$.
Here ${{a}_{1}}=10\text{ and }r=-\frac{1}{2}$.
Thus
${{S}_{n}}=\frac{10}{\left( 1+\frac{1}{2} \right)}$
Hence, the given statement is false.
For the statement to be true, ${{S}_{n}}=\frac{10}{\left( 1+\frac{1}{2} \right)}$.
