An geometric sequence or progression, is a sequence where each term is calculated by multiplying the previous term by a fixed number. Geometric SequencesThis fixed number is called the common ratio, r. The common ratio can be calculated by dividing any term by the one before it. The first term of a geometric sequence is shown by the variable a.
General Term, tn A geometric sequence can be written:
Geometric SeriesIf terms of a geometric sequence are added together a geometric series is formed. 2 + 4 + 8 + 16 is a finite geometric series To find the sum of the first n terms of a geometric sequence use the formula:
OR If the common ratio is a fraction i.e. -1 < r < 1 then an equivalent formula, shown below is easier to use.
Example What is the sum of the first 10 terms of the geometric sequence: 3, 6, 12, ...
The Sum to Infinity of a Geometric SequenceSpreadsheets are very useful for generating sequences and series. For a geometric sequence with a common ratio greater than 1:
It can be seen that as successive terms are added the sum of the terms increases. For a geometric sequence with a common ratio less than 1:
It can be seen that as successive terms are added the sum of the terms appears to be heading towards 16. This is called the sum to infinity of a geometric sequence and only applies when the common ratio is a fraction i.e. -1 < r < +1. The following formula can be used:
Example Find the sum to infinity of the geometric sequence 8, 4, 2, 1, ...
As can be seen from cell D10 in the spreadsheet above, 16 is the value the sums were heading towards. To see this concept clearly illustrated - Winona W. 1 Expert Answer The formula for a geometric sequence is the first term times r ^(n-1) where n is the number of terms. So it would be 2 * 3^(8-1) = 2* 3^7 = 4374 |