Question

So this question is from progression chapter of cengage book ​

Answer 1

Step-by-step explanation:

$S_{k} = \frac{k}{1 - \frac{1}{k + 1} }$

$\implies \: S_{k} = \frac{k(k + 1)}{k + 1 - 1}$

$\implies \: S_{k} = \frac{k(k + 1)}{k }$

$\implies \: S_{k} = (k + 1)$

Now,

$\sum^{10}_{k = 1} S_{k}$

$= \sum^{10}_{k = 1}(k + 1)$

$= \sum^{10}_{k = 1}(k) + \sum^{10}_{k = 1}(1)$

$= \frac{10(10 + 1)}{2} + 10$

$= \frac{10 \times 11}{2} + 10$

$= 55 + 10$

$= 65$