The sum to infinity of a geometric series is 108. The first term of the series is less than 40. The second term of the series is 24.

Find the first term and the common ratio of the series.

Show that the $n\,$ th term of the series, $u_n$ , can be written as $un=2n+13n−3u_n = \dfrac{2^{n+1}}{3^{n-3}}$ .

Hence show that $log⁡3un=n(log⁡32−1)+log⁡32+3\log_3 u_n = n\left(\log_3 2 - 1\right) + \log_3 2 + 3$ .

The sum to infinity of a geometric series is 108. The first term of the series is less than 40. The second term of the series is 24.

Find the first term and the common ratio of the series.

Show that the $n\,$ th term of the series, $u_n$ , can be written as $un=2n+13n−3u_n = \dfrac{2^{n+1}}{3^{n-3}}$ .

Hence show that $log⁡3un=n(log⁡32−1)+log⁡32+3\log_3 u_n = n\left(\log_3 2 - 1\right) + \log_3 2 + 3$ .

3.4 Sequences and Series (A-level only)

3.4 Sequences and Series (A-level only) Questions