Understanding Geometric Series

A geometric series is a sequence of terms which are multiplied or divided by the same amount each time.
The common multiplier is known as the common ratio, represented by ‘r’.
To qualify as a geometric series, all ratios between consecutive terms must be constant.

The Nth Term of a Geometric Series

The nth term of a geometric sequence is given by a*r^(n-1), where ‘a’ is the first term and ‘r’ is the common ratio.
Be able to use this formula, understanding what each part means and where they fit into the sequence.

Sum of Terms in a Geometric Series

The sum of the first ‘n’ terms in a geometric series can be calculated using the formula S_n = a*(1 - r^n) / (1 - r).
If the absolute value of r is less than 1, the series is convergent, and the sum to infinity is given by S_infinity = a / (1 - r).
If the absolute value of r is more than 1, the series does not converge and no sum to infinity exists.
Regardless of the value of r, the sum of the first ‘n’ terms can be calculated.

Geometric Series in Real Life Applications

Geometric series can be used to model growth and decay in the physical sciences, economics, and other areas.
Model problems using sequences, series and apply formulas to find nth terms or the sum of first n terms.

Remember the formulas and when to use each one - if you’re working in a finite sequence, use the first formula; if you’re working within an infinite sequence and looking for the sum of infinite terms, use the second, but check first whether the series converges or not.