Candidates should be able to:

• use the method of mathematical induction to establish a given result (e.g. \(\sum\limits_{r=1}^{n} r^4 = \frac{1}{4} n^2 (n+1)^2 , u_n = \frac{1}{2}(1 + 3^{n-1})\) for the sequence given by \(u_{n+1} = 3u_n – 1\) and \(u_1 = 1 , \left( \begin{array}{cc} 4 & -1 \\ 6 & -1 \end{array} \right)^n = \left( \begin{array}{cc} 3 \times 2^n -2 & 1-2^n \\ 3 \times 2^{n+1} – 6 & 3 – 2^{n+1} \end{array} \right) , 3^{2n} + 2 \times 5^n – 3\) is divisible by 8.);
• recognise situations where conjecture based on a limited trial followed by inductive proof is a useful strategy, and carry this out in simple cases. (e.g. find the \(n\)th derivative of \(xe^x\), find \(\sum\limits_{r=1}^{n} r \times r!\).

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Proof by induction – Further Pure Maths 1 Video (1:36:40)

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