Mathematics 7357: Sequences and Series Topical Past Paper Questions

Each of the series below shows the first four terms of a geometric series. Identify the only one of these geometric series that is convergent. [1 mark] Tick (✓) one box. $$ \begin{aligned}&0.1+0.2+0.4+0.8+\ldots\\&\begin{aligned}\\ &1-1+1-1+\ldots\\&\quad128-64+32-16+\ldots\\&\qu…

The $n$th term of a sequence is $u_{n}$. The sequence is defined by $$ u_{n+1}=p u_{n}+70 $$ where $u_{1}=400$ and $p$ is a constant. (a) Find an expression, in terms of p, for $ u_{2} $ [1 mark] It is given that $u_{3}=382$ (b) (i) Show that $p$ satisfies the equation 200p^2 + 3…

(a) A geometric sequence has first term 1 and common ratio $ \frac{1}{2} $ (a) (i) Find the sum to infinity of the sequence. [2 marks] (a) (ii) Hence, or otherwise, evaluate $ \sum_{n=1}^{\infty} (\sin 30^\circ)^n $ [2 marks] Find the smallest positive exact value of $\theta$, in…