Arithmetic Sequences: nth Term, Sum and Consecutive Terms

An arithmetic sequence is a sequence in which each term is obtained from the previous one by adding a constant difference d: a_n = a₁ + (n − 1)d. In brief: the sequence 5, 8, 11, 14, … has a₁ = 5 and d = 3, so a_n = 3n + 2. The sum of the first n terms is S_n = n(a₁ + a_n)/2, so S₁₀ = 10(5 + 32)/2 = 185. Each term is the average of its neighbours: a_n = (a_n−1 + a_n+1)/2.

Arithmetic sequences model situations with a constant increase: regular savings, steps of a staircase, evenly spaced seats. They are the discrete form of linear functions. This guide explains the definition, the formula for the nth term, the sum formula (with Gauss’s trick), the property of consecutive terms and how to recognise an arithmetic sequence, with a table of examples.

What is an arithmetic sequence?

A sequence (a_n) is arithmetic if the difference d = a_n+1 − a_n is the same for every n. If d > 0, the sequence is increasing, if d < 0 it is decreasing and if d = 0 it is constant. Examples: 2, 5, 8, 11, … (d = 3); 20, 17, 14, … (d = −3); 7, 7, 7, … (d = 0).

Because the terms change by a constant amount, the graph of the points (n, a_n) lies on a straight line with slope d.

What is the formula for the nth term?

Starting from a₁ and adding d each time, the nth term is a_n = a₁ + (n − 1)d. More generally, a_n = a_k + (n − k)d for any index k. Thus a_n = dn + (a₁ − d), a linear expression in n.

Example 1

a₁ = 7, d = 4. Then a_n = 7 + 4(n − 1) = 4n + 3. a₂₀ = 83.

Example 2: from two terms

a₃ = 11 and a₈ = 31. Five steps separate them, so 5d = 20 and d = 4. Then a₁ = 11 − 8 = 3, and a_n = 4n − 1.

Example 3: how many terms?

How many terms does 6, 10, 14, …, 98 have? 6 + 4(n − 1) = 98 gives n − 1 = 23 and n = 24.

What is the formula for the sum?

The sum S_n = a₁ + a₂ + … + a_n equals S_n = n(a₁ + a_n)/2 = n[2a₁ + (n − 1)d]/2.

The reasoning (a trick attributed to the young Gauss): write the sum forward and backward and add. Each pair a_k + a_n+1−k equals a₁ + a_n, and there are n pairs, so 2S_n = n(a₁ + a_n). For the numbers 1 to 100: S = 100 · 101/2 = 5,050. In general, 1 + 2 + … + n = n(n + 1)/2.

Example 4

Sum of 5, 8, 11, …, up to 14 terms: a₁₄ = 5 + 13 · 3 = 44, S₁₄ = 14 · (5 + 44)/2 = 343.

Example 5: sum of odd numbers

1 + 3 + 5 + … + (2n − 1) = n(1 + 2n − 1)/2 = n². For the first 12 odd numbers: 144.

Example 6: finding n from the sum

How many terms of 3, 7, 11, … sum to 465? S_n = n[6 + 4(n − 1)]/2 = n(2n + 1) = 465, so 2n² + n − 465 = 0. The discriminant is 1 + 3720 = 3721 = 61², so n = (−1 + 61)/4 = 15. Check: a₁₅ = 59, S = 15 · 62/2 = 465 ✓.

What is the property of consecutive terms?

In an arithmetic sequence every term except the first is the arithmetic mean of its two neighbours: a_n = (a_n−1 + a_n+1)/2. Equivalently, three numbers a, b, c are consecutive terms of an arithmetic sequence if and only if 2b = a + c. This gives a quick test and a way to find unknowns.

Example 7

For which x are x − 1, 2x + 1 and 5x − 2 consecutive terms of an arithmetic sequence? 2(2x + 1) = (x − 1) + (5x − 2), so 4x + 2 = 6x − 3 and x = 5/2. The terms are 1.5, 6, 10.5 with d = 4.5 ✓.

Example 8: inserting arithmetic means

Insert three numbers between 4 and 20 so that all five form an arithmetic sequence. 20 = 4 + 4d, so d = 4, and the numbers are 8, 12, 16.

Example 9: symmetric terms

Three consecutive terms have the sum 27 and the product 693. Let them be b − d, b, b + d: 3b = 27, so b = 9. The product 9(81 − d²) = 693 gives 81 − d² = 77, d = ±2. The terms are 7, 9, 11.

Worked examples

Example 10: a theatre

A theatre has 20 seats in the first row, and each next row has 2 more seats. In 25 rows: a₂₅ = 20 + 24 · 2 = 68 seats in the last row and S₂₅ = 25 · (20 + 68)/2 = 1,100 seats in total.

Example 11: savings

Someone saves 50 dollars in the first month and increases the amount by 10 dollars each month. After 12 months the last deposit is 160 and the total saved is 12 · 210/2 = 1,260 dollars.

Table of arithmetic sequences

Where does the sequence come from?

Arithmetic progressions occur in Babylonian and Egyptian mathematics: the Rhind papyrus has a problem about dividing bread in an arithmetic progression. The story that the ten-year-old Carl Friedrich Gauss summed the numbers from 1 to 100 within seconds by pairing them is well known, although its details are based on later retellings. The sum formula was known long before Gauss, however, including to Archimedes and Indian mathematicians such as Aryabhata.

Where is it used today?

Simple interest, steady-wage increases, linear depreciation and evenly spaced measurements are arithmetic sequences. The formulas also give the total of tiered structures, such as the number of bricks in a pyramid of rows 1, 2, 3, …, n (n(n + 1)/2).

Which mistakes are common?

Using n instead of n − 1

a_n = a₁ + (n − 1)d. The first term needs no steps.

Mixing up the number of terms and the number of steps

From a₃ to a₈ there are 5 steps, not 6.

Sign errors with a negative difference

For d = −3, a_n = a₁ − 3(n − 1). Use brackets in the substitution.

Using the sum formula for a non-arithmetic sequence

Check that the difference is constant before using S_n = n(a₁ + a_n)/2.