FP1 June 2014 (R) Q9
9.
(a) Prove by induction that, for \(n \in \mathbb{Z}^{+}\), \[\sum_{r=1}^{n} (r + 1)2^{r-1} = n2^n\] (5)
(b) A sequence of numbers is defined by \[u_1 = 0, \qquad u_2 = 32,\] \[u_{n+2} = 6u_{n+1} - 8u_n \qquad n \geqslant 1\] Prove by induction that, for \(n \in \mathbb{Z}^{+}\), \[u_n = 4^{n+1} - 2^{n+3}\] (7)
| Scheme | Marks |
|---|---|
| When n = 1, rhs = lhs = 2 | B1 |
| Assume true for \(n = k\) so \(\displaystyle\sum_{r=1}^{k} (r + 1)2^{r-1} = k2^k\) | |
| \(\displaystyle\sum_{r=1}^{k+1} (r + 1)2^{r-1} = k2^k + (k + 1 + 1)2^{k+1-1}\) M1: Attempt to add \((k + 1)^{\text{th}}\) term A1: Correct expression | M1A1 |
| \(= k2^k + (k + 2)2^k\) | |
| \(= 2 \times k2^k + 2 \times 2^k\) | |
| \(= (k + 1)2^{k+1}\) At least one correct intermediate step required. | A1 |
| If the result is true for \(\boldsymbol{n = k}\) then it has been shown true for \(\boldsymbol{n = k + 1}\). As it is true for \(\boldsymbol{n = 1}\) then it is true for all \(\boldsymbol{n}\) (positive integers.) cso, statements can be seen anywhere in the solution. | A1 |
Do not award final A if \(n\) defined incorrectly e.g. ‘\(n\) is an integer’ award A0 | |
| (5) |
| Scheme | Marks |
|---|---|
| When \(n = 1\) \(u_1 = 4^2 - 2^4 = 0\) \(4^2 - 2^4 = 0\) seen | B1 |
| When \(n = 2\) \(u_2 = 4^3 - 2^5 = 32\) \(4^3 - 2^5 = 32\) seen | B1 |
| True for \(n = 1\) and \(n = 2\) | |
| Assume \(u_k = 4^{k+1} - 2^{k+3}\) and \(u_{k+1} = 4^{k+2} - 2^{k+4}\) | |
| \(u_{k+2} = 6u_{k+1} - 8u_k\) \(= 6(4^{k+2} - 2^{k+4}) - 8(4^{k+1} - 2^{k+3})\) M1: Attempts \(u_{k+2}\) in terms of \(u_{k+1}\) and \(u_k\) A1: Correct expression | M1A1 |
| \(= 6.4^{k+2} - 6.2^{k+4} - 8.4^{k+1} + 8.2^{k+3}\) | |
| \(= 6.4^{k+2} - 3.2^{k+5} - 2.4^{k+2} + 2.2^{k+5}\) Attempt \(u_{k+2}\) in terms of \(4^{k+2}\) and \(2^{k+5}\) | M1 |
| \(= 4.4^{k+2} - 2^{k+5} = 4^{k+3} - 2^{k+5}\) | |
| So \(u_{k+2} = 4^{(k+2)+1} - 2^{(k+2)+3}\) Correct expression | A1 |
| If the result is true for \(\boldsymbol{n = k}\) and \(\boldsymbol{n = k + 1}\) then it has been shown true for \(\boldsymbol{n = k + 2}\). As it is true for \(\boldsymbol{n = 1}\) and \(\boldsymbol{n = 2}\) then it is true for all \(\boldsymbol{n}\) (positive integers.) cso, statements can be seen anywhere in the solution. | A1 |
Do not award final A if \(n\) defined incorrectly e.g. ‘\(n\) is an integer’ award A0 | |
| (7) | |
| (12 marks) |