Arithmetic Progression Calculator

Definition

A sequence is AP iff there exists a real number d such that for all n ≥ 2:

a_n - a_n-1 = d

General (nth) term

By induction, after n−1 steps from the first term:

a_n = a_1 + (n-1) d

Sum of the first n terms

Pair the first and last, the second and penultimate, and so on. Each pair sums to a₁ + aₙ, and there are n/2 such pairs. The 'odd-middle' case gives the same expression:

S_n = (n)/(2)\bigl(2 a_1 + (n-1) d\bigr) = (n)/(2)(a_1 + a_n)

Mean of an arithmetic sequence

The arithmetic mean of all n terms is (a₁ + aₙ)/2, which is the value of the middle term when n is odd and the average of the two central terms when n is even.

AP vs GP

In an AP, terms are added (constant difference); in a geometric progression (GP), they are multiplied (constant ratio). AP linear growth, GP exponential growth — these are the two foundational linear-vs-exponential structures of algebra.

Two Ways to Write the Sum

The sum of the first n terms has two equivalent forms, and choosing the right one saves work. When you know the first and last terms, use S_n = \tfracn2(a_1 + a_n) — the number of terms times the average of the ends. When you know the first term and the common difference but not the last, substitute a_n = a_1 + (n-1)d to get the expanded form.

The first version carries the intuition. Gauss's schoolroom trick of summing 1 to 100 pairs the first with the last, the second with the second-to-last, and so on: every pair totals 101, and there are 50 pairs, giving 5050. That is exactly \tfracn2(a_1 + a_n). Because the terms are evenly spaced, their mean is simply the average of the two endpoints — which is not true for most sequences and is precisely what makes arithmetic progressions so tractable.

S_n = (n)/(2)(a_1 + a_n) = (n)/(2)[2a_1 + (n-1)d]

Finding n, d, or a Missing Term

The four quantities a_1, d, n and a_n are linked by one equation, so any three determine the fourth. Rearranging gives d = (a_n - a_1)/(n-1) and n = (a_n - a_1)/d + 1.

Note the n-1 rather than n: there are n terms but only n-1 gaps between them, and off-by-one errors here are the most common mistake in the whole topic. A quick check: the sequence 3, 7, 11, 15 has 4 terms and 3 gaps of 4, so a_4 = 3 + 3(4) = 15. If a computed n comes out non-integer, the target value simply is not a member of the progression — useful in itself when testing whether a given number appears in a sequence.

The arithmetic mean of two numbers is the term that sits between them in an AP, which is where the name comes from; inserting k such means between two values is a standard exercise solved by treating the result as an AP with k+2 terms.

a_n = a_1 + (n-1)d, \qquad d = (a_n - a_1)/(n-1)

Where Arithmetic Progressions Appear

Any process that adds a fixed amount each period is an AP. Simple interest grows the balance by the same absolute sum every year, unlike compound interest which is geometric. Straight-line depreciation writes off an identical amount annually. Seating in an auditorium that gains two seats per row, stacked pipes in a triangular pile, and the total distance fallen in successive seconds under constant gravity are all arithmetic.

The last is worth noting: an object in free fall covers distances in the ratio 1 : 3 : 5 : 7 during equal successive time intervals — an AP with d = 2 — which was Galileo's route to the law of falling bodies before calculus existed. Recognising an AP in a word problem usually means spotting the phrase "increases by the same amount each time"; if instead it increases by the same *percentage*, the sequence is geometric and these formulas do not apply.

Worked Example

Inputs

• firstTerm: 1
• commonDiff: 2
• numTerms: 5

Result: lastTerm: 9 sum: 25 mean: 5

Explanation

a₁ = 1, d = 2, n = 5: terms are 1, 3, 5, 7, 9. Last term a₅ = 1 + 4·2 = 9. Sum S₅ = 5/2 · (1 + 9) = 25. Mean = 5.

Second Scenario

Inputs

• firstTerm: 2.25
• commonDiff: 2
• numTerms: 5

Result: lastTerm: 9 sum: 25 mean: 5

Explanation

This scenario uses different inputs (firstTerm = 2.25, commonDiff = 2, numTerms = 5) to show how changing one variable affects the arithmetic progression result. Run the calculator above with these values to get the exact updated output with step-by-step work.