1.26: Past-paper question 6

(i) A geometric series $\displaystyle G$ has first term $\displaystyle (3 + \sqrt{3})$ and common ratio $\displaystyle r$

The second term of $\displaystyle G$ is $\displaystyle \sqrt{3}$

(a) Find, showing all your working, the exact value of $\displaystyle r$

Give your answer in the form $\displaystyle \frac{p + \sqrt{3}}{q}$ where $\displaystyle p$ and $\displaystyle q$ are integers to be found.

(b) Explain why $\displaystyle G$ is convergent.

(ii) A different geometric series $\displaystyle H$ has first term 8 and common ratio 0.6

The sum to $\displaystyle n$ terms of $\displaystyle H$ is $\displaystyle S_n$ and the sum to infinity of $\displaystyle H$ is $\displaystyle T$

Find, using logarithms, the least value of $\displaystyle n$ such that $\displaystyle T - S_n < 0.12$