OCR A-Level Further Maths A series and proof overview quiz quiz

assume the result for $n = k$

verify a base case

prove the result for $n = k + 1$

write the conclusion

$n = k + 1$

all $n$

$n = k$

$n = 0$

it is optional and can be left out without losing any marks

it simply repeats the base case that was already verified earlier

it is what actually proves the inductive hypothesis is true

it earns a mark by stating the result holds for all $n$

$\frac{1}{2}n(n+1)$

$n(n+1)$

$\frac{1}{6}n(n+1)(2n+1)$

$n^2$

$\frac{1}{6}n(n+1)(2n+1)$

$\frac{1}{4}n^2(n+1)^2$

$\frac{1}{2}n(n+1)$

$n^3$

$c$

$\frac{1}{2}cn$

$cn$

$c^n$

a product of consecutive terms

a sum of squares

a constant

a difference $f(r) - f(r+1)$

partial fractions

differentiation

integration

the binomial theorem

$-\frac{b}{a}$

$\frac{c}{a}$

$\frac{b}{a}$

$-\frac{c}{a}$

$\frac{b}{a}$

$-\frac{d}{a}$

$-\frac{b}{a}$

$\frac{c}{a}$

$(\alpha + \beta)^2$

$(\alpha + \beta)^2 + 2\alpha\beta$

$\alpha\beta - (\alpha + \beta)$

$(\alpha + \beta)^2 - 2\alpha\beta$

$x \to x - 2$

$x \to x + 2$

$x \to 2x$

$x \to \frac{x}{2}$