Proof – Edexcel FP1

6. Proof – Edexcel Further Pure Mathematics 1 (FP1)

What students need to learn:
Proof by mathematical induction.
To include induction proofs for
(i) summation of series
eg show \(\displaystyle\sum_{r=1}^n r^3 = \frac{1}{4} n^2(n + 1)^2\) or
\(\displaystyle\sum_{r=1}^n r(r + 1) = \dfrac{n(n + 1)(n + 2)}{3}\)

(ii) divisibility
eg show \(3^{2n} + 11\) is divisible by 4.

(iii) finding general terms in a sequence
eg if \(u_{n+1} = 3u_n + 4\) with
\(u_1 = 1\) , prove that \(u_n = 3^n – 2\) .

(iv) matrix products
eg show
\(\begin{pmatrix} -2& -1\\ 9& 4\end{pmatrix}^n = \begin{pmatrix} {1-3n}& -n\\ 9n& {3n+1} \end{pmatrix}\) .


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