Arithmetic sequence nth term:
a + (n − 1)d, step by step

What makes a sequence arithmetic

In an arithmetic sequence, each term is found by adding the same fixed number to the one before. That number is the common difference, d. The list 5, 8, 11, 14, 17 has d = 3. The list 100, 95, 90, 85 has d = −5, so it counts downwards.

To check, subtract each term from the next. If the answer is always the same, you have an arithmetic sequence. If it changes, the sequence is something else, perhaps geometric or quadratic.

The difference can be a fraction or negative or zero. A sequence with d = 0 is just the same number repeated, which still satisfies the definition.

The nth term formula

The n − 1 is the part people question. The first term has had no steps added to it, the second term has had one, and the nth has had n − 1. If you used n × d instead, the first term would already include a step and everything would be shifted.

Writing out the list to confirm is not necessary, but a quick check is: the 20th term is 19 steps beyond the first, so 19 × 3 = 57 more than 5.

Many exam and spreadsheet problems give two non-adjacent terms instead of the first term and the difference. Suppose the 4th term is 14 and the 9th is 39. Five steps separate them, so d = (39 − 14) ÷ 5 = 5, and the first term is 14 − 3 × 5 = −1. Always find d first by dividing the change in value by the change in position, then step back to the first term.

Finding which term a number is

Run the formula backwards when you know a term and want its position. Rearranged, n = (aₙ − a) ÷ d + 1.

If n is not a whole number, the target is not in the sequence. Try 100 in the same list: (100 − 5) ÷ 3 = 31.67, so 100 does not appear.

Adding up the first n terms

The formula works because the first and last terms, the second and second-last, and so on, all pair up to the same total. Continuing the 5, 8, 11 example: the sum of the first 20 terms is 20 ÷ 2 × (5 + 62) = 10 × 67 = 670.

A realistic case: a growing monthly saving

You start a recurring deposit of ₹2,000 in month 1 and raise it by ₹250 every month. What is the deposit in month 24, and how much have you put in by then?

This ignores interest. A deposit that earns compound interest is a different problem, closer to a geometric one.

Rows of seats and flights of steps

Patterns of rows are a natural place to see this at work. A hall has 20 seats in the first row and 2 more in each following row, over 15 rows. The last row has 20 + 14 × 2 = 48 seats, and the hall seats 15 ÷ 2 × (20 + 48) = 510 people.

Stair risers, scaffolding levels, pipe stacks, brick courses in a triangular wall and the markings on a ruler are all arithmetic. The common thread is a fixed step. If any step is not the same, perhaps because a landing intervenes, split the problem at that point and treat each part as its own sequence.

The link between sequences and straight lines

Plot position n on the horizontal axis and the term aₙ on the vertical axis and the points fall on a straight line with slope d. Expanding the formula shows this: aₙ = dn + (a − d). The common difference is the slope, and a − d is the value that the line would take at position 0.

For 5, 8, 11, … that line is aₙ = 3n + 2. Check: n = 20 gives 60 + 2 = 62, as before. This rewritten form is quicker when you need many terms, since each is one multiplication and one addition.

Seen this way, an arithmetic sequence is a linear function evaluated only at whole numbers, which is why the same ideas about rate of change carry over.

Decreasing sequences and checks

A negative d works in exactly the same way. Starting at 100 and dropping 5 each time, the 12th term is 100 + 11 × (−5) = 45, and the first twelve terms sum to 12 ÷ 2 × (100 + 45) = 870.

• Always confirm d from two pairs of terms before using the formula.
• Remember that n counts from 1. If the problem starts at term 0, adjust.
• Keep the sign of d when it is negative; write (n − 1) × (−5) with brackets.
• If the sequence is quadratic, such as 1, 4, 9, 16, the first differences are not constant and this formula does not apply.