FP1 January 2010 Q3
3. A sequence of numbers is defined by \[\begin{aligned} u_1 &= 2, \\ u_{n+1} &= 5u_n - 4, \qquad n \geqslant 1. \end{aligned}\]
Prove by induction that, for \(n \in \mathbb{Z}^+\), \(u_n = 5^{n-1} + 1\). (4)
| Scheme | Marks |
|---|---|
| For \(n = 1\): \(u_1 = 2\), \(u_1 = 5^0 + 1 = 2\) | B1 |
| Assume true for \(n = k\): | |
| \(u_{k+1} = 5u_k - 4 = 5(5^{k-1} + 1) - 4 = 5^k + 5 - 4 = 5^k + 1\) | M1 A1 |
| \(\therefore\) True for \(n = k + 1\) if true for \(n = k\). True for \(n = 1\), \(\therefore\) true for all \(n\). | A1 cso |
| [4] |
Notes
Accept \(u_1 = 1 + 1 = 2\) or above B1
\(5(5^{k-1} + 1) - 4\) seen award M1
\(5^k + 1\) or \(5^{(k+1)-1} + 1\) award first A1
All three elements stated somewhere in the solution award final A1