# Proof by induction – Further Pure Maths 1

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 $$\sum\limits_{r=1}^{n} r \times r!$$.

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