Sequence Sum Calculator

What Is a Sequence Sum?

A sequence is an ordered list of numbers: 2, 5, 8, 11, 14. A series is what you get when you add those numbers: 2 + 5 + 8 + 11 + 14 = 40. The sequence sum, or partial sum Sₙ, is the total of the first n terms.

The two ideas are easy to mix up. “What is the 5th term?” is a sequence question (answer: 14). “What is the total of the first 5 terms?” is a series question (answer: 40).

How to Find the Sum of an Arithmetic Sequence

An arithmetic sequence adds the same number d each time. Identify the first term a₁, the common difference d and the number of terms n. The last term is aₙ = a₁ + (n − 1)d, and the sum is Sₙ = n(a₁ + aₙ)/2, or equivalently Sₙ = n/2 [2a₁ + (n − 1)d].

For 2, 5, 8, 11, 14: a₁ = 2, d = 3, n = 5 and a₅ = 14, so S₅ = 5(2 + 14)/2 = 40. The formula works because pairing the first and last terms, the second and second-to-last, and so on, gives n/2 pairs with the same total a₁ + aₙ. This is the trick Gauss is said to have used to add 1 + 2 + … + 100 = 5,050.

How to Find the Sum of a Geometric Sequence

A geometric sequence multiplies by the same number r each time. For r ≠ 1 the sum of the first n terms is Sₙ = a₁(1 − rⁿ)/(1 − r), which can also be written a₁(rⁿ − 1)/(r − 1).

For 2, 6, 18, 54, 162: a₁ = 2, r = 3, n = 5, so S₅ = 2(3⁵ − 1)/(3 − 1) = 2 · 242/2 = 242. When r = 1 every term equals a₁, so Sₙ = n · a₁; the general formula cannot be used then, because it would divide by 0.

Sum From Term m to Term n

To add a block of terms in the middle of a sequence, find the first and last terms of the block, aₘ and aₙ, and count the terms: n − m + 1, since both ends are included. For an arithmetic sequence the block sum is (n − m + 1)(aₘ + aₙ)/2. For 2, 5, 8, 11, 14, 17, …, terms 3 to 6 are 8 + 11 + 14 + 17 = 4(8 + 17)/2 = 50.

For a geometric sequence the block is itself geometric, starting at aₘ with the same ratio, so the finite sum formula applies with aₘ as the first term and n − m + 1 terms.

How to Find the Sum of an Infinite Geometric Series

An infinite series adds infinitely many terms. For a geometric series a₁ + a₁r + a₁r² + … the partial sums Sₙ = a₁(1 − rⁿ)/(1 − r) settle down only if rⁿ → 0, which happens exactly when |r| < 1. Then the infinite sum is S∞ = a₁/(1 − r). For 10 + 5 + 2.5 + 1.25 + …, r = 1/2 and S∞ = 10/(1 − 1/2) = 20.

When |r| ≥ 1 the series diverges and has no finite sum. With r = 2 the terms grow without bound, with r = 1 the total keeps growing by a₁, and with r = −1 the partial sums jump between a₁ and 0 forever. If r = 0 the series is a₁ + 0 + 0 + …, whose sum is a₁.

Sequence vs Series

A sequence is the list: 3, 6, 12, 24, … A series is the sum of that list: 3 + 6 + 12 + 24 + … A finite series stops after n terms and always has a value. An infinite series goes on forever, and its value, if it has one, is the limit of the partial sums S₁, S₂, S₃, …

Arithmetic Sequence vs Geometric Sequence

In an arithmetic sequence the difference between consecutive terms is constant (2, 5, 8, 11 has d = 3), so the terms change linearly and an infinite arithmetic series never has a finite sum unless every term is 0. In a geometric sequence the ratio between consecutive terms is constant (2, 6, 18, 54 has r = 3), so the terms change exponentially and an infinite series converges when |r| < 1.

To tell which kind of list you have, compare the differences and the ratios of consecutive terms. If neither is constant, as in 2, 4, 7, 11, the list is neither; you can still add it term by term.

Finite vs Infinite Series

A finite series such as S₁₀ always has a definite value, which you find with the formulas above or by adding the terms. An infinite series only has a value if its partial sums approach a limit. Among geometric series that happens exactly when |r| < 1. The infinite sum is never simply “the last term times something”: it is the number the running totals get closer and closer to.

Real-World Uses of Sequence Sums

Savings that grow by a fixed amount: saving $50 in month 1 and $10 more each month, the first 12 months total 12(50 + 160)/2 = $1,260 (an arithmetic series).

Regular deposits with interest: the value of equal yearly deposits with compound growth is a geometric series with r = 1 + interest rate.

Depreciation: equipment that keeps 80% of its value each year follows a geometric sequence with r = 0.8.

Physics: a ball that bounces back to 60% of its previous height travels a total distance given by an infinite geometric series with r = 0.6.

Computer science: doubling work at each step (1 + 2 + 4 + … + 2ⁿ⁻¹ = 2ⁿ − 1) and the cost of repeated halving are geometric sums.

Worked Examples

Arithmetic: a₁ = 20, d = −3, n = 5 gives 20 + 17 + 14 + 11 + 8 = 5(20 + 8)/2 = 70.

Constant: a₁ = 5, d = 0, n = 10 gives 10 × 5 = 50.

Geometric with r = 1: a₁ = 7, n = 10 gives 10 × 7 = 70.

Negative ratio: a₁ = 8, r = −1/2, n = 4 gives 8 − 4 + 2 − 1 = 8(1 − 1/16)/(3/2) = 5.

List: 2, 4, 6, 8, 10 adds to 30 and is arithmetic with d = 2; 2, 4, 7, 11 adds to 24 and is neither arithmetic nor geometric.