Arithmetic Progression

Definition

An arithmetic progression (AP) is a sequence in which, starting from the second term, each term is obtained by adding the same number d to the previous one. d is called the common difference: d = aₙ₊₁ − aₙ.

• If d > 0 the progression increases, if d < 0 it decreases, if d = 0 all terms are equal.
• Example: 3, 7, 11, 15, ... where a₁ = 3, d = 4.

Main formulas

What is found	Formula
n-th term	aₙ = a₁ + (n − 1) · d
Sum of the first n terms	Sₙ = (a₁ + aₙ) · n / 2
Sum (second form)	Sₙ = (2a₁ + (n − 1) · d) · n / 2
Middle term	aₙ = (aₙ₋₁ + aₙ₊₁) / 2

• If the sums of the indices of the terms are equal, the sums of the terms are also equal: if k + m = p + s, then aₖ + aₘ = aₚ + aₛ.
• If three consecutive numbers a, b, c form an AP, then 2b = a + c.
• aₙ = aₘ + (n − m) · d. If two terms are known, d = (aₙ − aₘ) / (n − m).

Solution procedure

1. Find a₁ and d from the given data
2. Write down the formula for the n-th term or the sum
3. Find n from the equation (it must be a positive integer)
4. Check the answer using the numbers

Examples

• If a₁ = 3, d = 4, then a₁₀ = 3 + 9 · 4 = 39. S₁₀ = (3 + 39) · 10 / 2 = 210.
• If a₃ = 7, a₇ = 19, then d = (19 − 7) / 4 = 3, a₁ = 7 − 2 · 3 = 1.
• Multiples of 5 from 5 to 50: 5, 10, ..., 50, that is 10 terms, the sum is (5 + 50) · 10 / 2 = 275.

Additional notes

• The sum of the natural numbers from 1 to n is n(n + 1)/2.
• Number of terms = (last − first)/d + 1
• In the sum formula, do not forget to divide by 2.