Solve simple summation problems

Summation Basics

Understand the meaning of a sum from mathematics, where the total result is obtained by adding several numbers, values or elements together.
Familiarise yourself with the sigma notation (Σ), which represents summation. Each term in the sum corresponds to a term in the series from a starting point (i.e., the lower limit) to an ending point (i.e., the upper limit).
Distinguish between finite sums and infinite series. Finite sums have a definite number of terms whereas infinite series continue indefinitely.

Common Summation Formulas

The sum of the first ‘n’ natural numbers is given by the formula n(n+1)/2.
The sum of the squares of the first ‘n’ natural numbers is given by the formula n(n+1)(2n+1)/6.
The sum of the cubes of the first ‘n’ natural numbers is given by the formula [n(n+1)/2]^2.

Working with Sequences and Series

Recognise that sequences are ordered lists of numbers where each term follows a certain pattern.
Understand that a series is merely the sum of the terms of a sequence.
Distinguish between arithmetic, geometric and harmonic series. Each has a different pattern for determining the subsequent numbers.
For arithmetic series, the difference between any two successive terms is constant. If ‘a’ is the first term and ‘d’ is the common difference, the nth term formula is a + (n-1)d and the sum of an arithmetic sequence (S) is n/2(2a + (n-1)d).
For geometric sequences, any term divided by its preceding term gives a constant ratio. If ‘a’ is the first term and ‘r’ is the common ratio, the nth term formula is ar^(n-1) and the sum (S) of a geometric series is either a(r^n - 1)/(r - 1) if r>1 or a(1 - r^n)/(1 - r) if r<1 for finite series.
Harmonic series is a type of divergent infinite series where the nth term is 1/n.

Transforming sums into functions

Understand that we can turn series into generating functions - a way to encode an infinite sequence of numbers by treating them as the coefficients of a power series.
Realise how solving problems related to these generating functions can help in finding the sum of the series.