A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. The common ratio (r) is obtained by dividing any term by the preceding term, i.e.,
where | r | common ratio | | a1 | first term | | a2 | second term | | a3 | third term | | an-1 | the term before the n th term | | an | the n th term |
The geometric sequence is sometimes called the geometric progression or GP, for short.
For example, the sequence 1, 3, 9, 27, 81 is a geometric sequence. Note that after the first term, the next term is obtained by multiplying the preceding element by 3.
The geometric sequence has its sequence formation:
To find the nth term of a geometric sequence we use the formula:
where | r | common ratio | | a1 | first term | | an-1 | the term before the n th term | | n | number of terms |
Sum of Terms in a Geometric Progression
Finding the sum of terms in a geometric progression is easily obtained by applying the formulas: nth partial sum of a geometric sequence
sum to infinity
where | Sn | sum of GP with n terms | | S∞ | sum of GP with infinitely many terms | | a1 | the first term | | r | common ratio | | n | number of terms |
Examples of Common Problems to Solve
Write down a specific term in a Geometric Progression
Write down the 8th term in the Geometric Progression 1, 3, 9, ...
Finding the number of terms in a Geometric Progression
Find the number of terms in the geometric progression 6, 12, 24, ..., 1536
Finding the sum of a Geometric Series
Find the sum of each of the geometric series | |
Finding the sum of a Geometric Series to Infinity
Converting a Recurring Decimal to a Fraction
Decimals that occurs in repetition infinitely or are