1.4: Sum of an arithmetic series in logarithmic form

4PM1/1/June/2021 — Question 4 · 5 marks

The n\displaystyle nth term of an arithmetic series is un\displaystyle u_n, where
un=(n+1)ln 4.u_n=(n+1)\mathrm{ln}\,4.
Given that the sum of the first n\displaystyle n terms of the series is Sn\displaystyle S_n, show that
Sn=ln 2(n2+an),S_n=\mathrm{ln}\,2^{\left(n^2+an\right)},
where a\displaystyle a is an integer whose value is to be found.
(5)