Geometric sequences calculator
This tool can help you find nth term and the sum of the first n terms of a geometric progression. Also, this calculator can be used to solve more complicated problems. For example, the calculator can find the first term (a1) and common ratio (r) if a2=6 and a5=48. The calculator will generate all the work with detailed explanation.
Tutorial
What is geometric sequence?
Geometric sequence is a list of numbers where each term is obtained by multiplying the previous term by a constant. The constant is called the common ratio (r).
Example 1: The sequence of numbers 2, 6, 18, 54,... is geometric because the ratio between any two consecutive numbers is equal to 3.
Example 2: The sequence 3, 6, 12, 20,... is not geometric because the ratio between terms is not constant.
Geometric sequence notation
Example 3: In the sequence 2, 6, 18, 54,... we have a1 = 2, r = 3, a3= 18, S4 = 80.
Geometric sequence formulas
Worked examples
Example 4. Find the 10-th element of a sequence 5, 15, 45, 135, 405.
The first term is a = 5, the common ratio is r = 3. We will use rule for any term an = a1·rn-1.
a10 = a1·r10-1=5·39 = 98 415
Example 5. Find the next term of the sequence 4, 20, 100, 500
The common ratio of the sequence is r = 5, the last element is 500. The next element is 500 · 5 = 2500
Example 6. Find the sum of the first 7 elements of a sequence {1, 5, 25, 125, ... }
The common ratio of the sequence is r = 5, the first element is a1 = 1. We will use rule for sum of first n terms.
Sn = a(rn-1)/(r-1)
S10 = 1(510-1)/(5-1)
S10 = 9 765 624/4
S10 = 9 765 624/4
S10 = 2 441 406