LifeTally
Geometric sequence sum and last term
What are the last term, sum and arithmetic mean when you multiply each term by the same factor?
Enter a signed first term, a constant multiplier and 1 to 1000 terms. Negative factors, zeros and decimal fractions are supported.
Your finite sequence
r is a multiplier, not a percentage change. A negative r alternates the signs of nonzero terms. n includes the first term.
The 1000-term maximum is this tool's processing limit, not a mathematical limit or a recommended count.
How the estimate works
The first term is a. Each following term is the preceding term multiplied by r. Add exactly the first n terms; divide that sum by n for the arithmetic mean. For r = 1, the sum is n × a.
t(1) = a; t(k + 1) = r × t(k)
last = a × r^(n − 1) (n = 1: a)
S = a × (1 − r^n) ÷ (1 − r) (r ≠ 1)
S = n × a (r = 1); mean = S ÷ nThe multiplier is supplied, not inferred by dividing terms. With r = 0, every term after the first is zero; with a = 0 and a valid multiplier, every term is zero.
One finite sequence with a constant supplied multiplier. This does not infer a rule from a list, sum an infinite series, test convergence, find a geometric or weighted mean, convert units or forecast financial outcomes. Terms must use one consistent value or unit basis; r is unitless.
OpenStax: geometric sequences ↗OpenStax: finite geometric sums ↗
Check the filled example
Our example: a = 3, r = −2, n = 5 gives 3 → −6 → 12 → −24 → 48. Last term 48, sum 33, arithmetic mean 33 ÷ 5 = 6.6. These are chosen mathematical inputs, not a real-world forecast.
Accepted input values
- First term a
- 3
- Multiplier r
- -2
- Number of terms n
- 5
Plan the next step
If the same amount is added at every step, use an arithmetic progression instead.
Arithmetic progression →