A geometric series is $a + ar + ar^2 + ...$ (a) Prove that the sum of the first $n$ terms of this series is given by $S_n = \frac{a(1 - r^n)}{1 - r}$ - Edexcel - A-Level Maths Pure - Question 2 - 2006 - Paper 2

Using the formula for the sum of a geometric series:

$S_n = \frac{a(1 - r^n)}{1 - r}$

Here, $a = 100(2^1) = 200$, $r = 2$, and $n = 10$.

We can substitute these values into the formula:

$S_{10} = \frac{200(1 - 2^{10})}{1 - 2} = 200(2^{10} - 1) = 200(1024 - 1) = 200 \times 1023 = 204600.$

For the series to converge, the common ratio must satisfy $|r| < 1$.

In this case, the first term $a = \frac{5}{6}$ and the common ratio $r = \frac{1}{3}$.

The sum to infinity is given by:

$S = \frac{a}{1 - r} = \frac{\frac{5}{6}}{1 - \frac{1}{3}} = \frac{\frac{5}{6}}{\frac{2}{3}} = \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4}.$

The condition for convergence of an infinite geometric series is:

$|r| < 1.$

This means that the absolute value of the common ratio must be less than 1.