Sequences & Series

(a)Find an expression for $\displaystyle \sum_{r=1}^{n} (r^2 + 3r)$ in terms of $\displaystyle n.$
Express your answer in the form $\displaystyle \frac{1}{3}n(n + a)(n + b).$

(b)Hence, or otherwise, find $\displaystyle \sum_{r=11}^{20} (r^2 + 3r).$

The fourth and seventh terms of a geometric sequence are 9 and 243 respectively.

(a)Find the:
(i) common ratio
(ii) first term.

(b)Show that $\displaystyle \frac{S_{2n}}{S_n} = 1+3^n$ where $\displaystyle S_n$ represents the sum of the first $\displaystyle n$ terms of this geometric sequence.

A geometric sequence of positive terms has third term 36 and fifth term 16.

(a)Calculate the value of the common ratio.

(b)Calculate the value of the first term.

(c)State why the associated geometric series has a sum to infinity.

(d)Find the value of this sum to infinity.

An arithmetic sequence has first term $\displaystyle -3$ and common difference $\displaystyle d.$

(a)State an expression for the third term.

The eighth term is five times the third term.

(b)Find the value of $\displaystyle d.$

(c)Determine algebraically the least number of terms required so that the sum of the associated series is greater than 500.

Find and fully factorise an expression for $\displaystyle \sum_{r=1}^{n} (r^3 - 3r).$