Enter the first few terms – 3, 7, 11, 15 – to find the rule, the nth term formula, the next terms and the sum, or evaluate a sum in sigma notation such as Σ (2k + 1) from k = 1 to 10. Geometric series also get their sum to infinity.
Arithmetic and geometric sequences
| Formula | Arithmetic | Geometric |
|---|---|---|
| Pattern | Add the same number d | Multiply by the same number r |
| Test | Differences are equal | Ratios are equal |
| nth term | a_n = a_1 + (n - 1)d | a_n = a_1 r^{n - 1} |
| Sum of n terms | S_n = \frac{n}{2}(a_1 + a_n) | S_n = \frac{a_1(1 - r^n)}{1 - r} |
| Sum to infinity | none (unless every term is 0) | S_\infty = \frac{a_1}{1 - r} when |r| < 1 |
Example: the 20th term of 3, 7, 11, 15, …
d = 4,\quad a_{20} = 3 + 19 \cdot 4 = 79Example: the 10th term and the sum of the first 10 terms of 2, 6, 18, 54, …
r = 3,\quad a_{10} = 2 \cdot 3^{9} = 39{,}366,\quad S_{10} = \frac{2(1 - 3^{10})}{1 - 3} = 59{,}048Open the 10th term or the sum in the calculator. A geometric sequence grows fast: the 10th term is already almost 20,000 times the first.
Which term is it? Working backwards
To find where a number appears, set the nth-term formula equal to it and solve for n. In 3, 7, 11, 15, … the rule is a_n = 4n - 1, so 99 is the term where 4n − 1 = 99: n = 25, the 25th term. If n comes out as a fraction or a negative number, the number is not in the sequence – 50 would need n = 12.75, so it never appears.
Sequences in everyday sums
- Saving a little more each month is arithmetic: ₹500, ₹600, ₹700, … for 12 months adds up to
\frac{12}{2}(500 + 1600) = 12{,}600. - A pay rise of 5% a year is geometric with r = 1.05: a salary of ₹6 lakh becomes
6 \times 1.05^{9} \approx 9.31lakh in the 10th year. - A bouncing ball that rises to 3/4 of its height each time travels a geometric series – its total distance is finite even though it bounces for ever.
Other patterns
- Quadratic – when the second differences are equal (1, 4, 9, 16, …), the nth term is
An^2 + Bn + Cwith2Aequal to the second difference. For 2, 5, 10, 17 the differences are 3, 5, 7 and the second difference is 2, so A = 1 and the rule is n² + 1. - Fibonacci-type – each term is the sum of the two before it (1, 1, 2, 3, 5, …).
Sums in sigma notation
The calculator counts the terms, then uses the quickest formula: the arithmetic or geometric series formula, or the power sums
\sum_{k=1}^{n} k = \frac{n(n + 1)}{2},\qquad \sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6},\qquad \sum_{k=1}^{n} k^3 = \left(\frac{n(n + 1)}{2}\right)^2A sum that does not start at 1 is the sum up to the top limit minus the sum below the bottom limit.
Frequently asked questions
When does an infinite geometric series have a sum?
Only when the common ratio is between −1 and 1. Then the sum is a₁/(1 − r); for 8 + 4 + 2 + 1 + … it is 8/(1 − ½) = 16.
How many terms do I need to type?
Three terms are enough for arithmetic and geometric sequences; give four or more for a quadratic or Fibonacci-type pattern.