A sequence ${{u}_{0}}$, ${{u}_{1}}$, ${{u}_{2}}$… is given by ${{u}_{0}}=\frac{3}{2}$ and \(

Question

A sequence ${{u}_{0}}$, ${{u}_{1}}$, ${{u}_{2}}$… is given by ${{u}_{0}}=\frac{3}{2}$ and ${{u}_{n}}={{u}_{n-1}}+{{2}^{n}}-n$ for $n\ge 1$. Find ${{u}_{1}}$, ${{u}_{2}}$ and ${{u}_{3}}$.[3] By considering $\sum\limits_{r=1}^{n}{\left( {{u}_{r}}-{{u}_{r-1}} \right)}$, find a formula for ${{u}_{n}}$ in terms of $n$.[5]

Accepted Answer

(i) ${u_1 = \frac{5}{2}, u_2 = \frac{9}{2}, u_3 = \frac{19}{2}}$ (ii) ${u_n = 2^{n+1} - \frac{1}{2} - \frac{n(n+1)}{2}}$

2017 PJC P1 Q2

2017 PJC P1 Q2

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