The principle of mathematical induction involves:
Much like dominoes falling, the proof for
Series and induction are covered at Further Maths Pure. This builds on existing knowledge of Sequences and Series from single Maths, and is further developed in Further Sequences and Series in Further Maths Additional Pure.
The following formulae can be used without proof, where the second and third formulae are given in the formula book:
Series can be manipulated in the following ways:
If the general term of a series can be written as
The series won't always take this exact form; sometimes cancellations can happen two terms apart. This form can often be obtained using partial fractions. When using the method of differences, make it clear what the cancellation pattern is.
Find a formula for the
We start by writing out the first terms of the series to try and spot a pattern. Each term contains a negated element from the previous one:
When summing these terms, the matched pairs cancel, leaving only the terms in white, so