Arithmetic Sequence Formula

It finds any term in a sequence that goes up (or down) by the same fixed step each time.

a_n = a_1 + (n - 1)d

Each term is one fixed step d above the previous one. The formula jumps straight to any term.

What each symbol means

• a_n: the term you want (for example the 10th term)
• a_1: the first term of the sequence
• n: the position of the term you want (1st, 2nd, 3rd, and so on)
• d: the common difference (the fixed step from one term to the next)

An arithmetic sequence is a list of numbers that changes by the same amount every step. Like 3, 7, 11, 15, and so on, going up by 4 each time. That fixed step is called the common difference.

You could keep adding 4 over and over to reach the 50th term, but that is slow. The formula a_n = a_1 + (n - 1)d jumps straight to any term you want.

Why (n minus 1)?: The first term needs no steps. To reach the 2nd term you take 1 step, the 3rd term 2 steps, and so on. So reaching the nth term takes (n - 1) steps. That is why the formula uses (n - 1), not n.

A worked example

Find the 10th term of 3, 7, 11, 15, ...

1. Identify the parts: first term a_1 = 3, common difference d = 4 (each step adds 4), and n = 10.
2. Write the formula: a_n = a_1 + (n - 1)d.
3. Put in the numbers: a_10 = 3 + (10 - 1) × 4.
4. Work the bracket: 10 - 1 = 9.
5. Multiply: 9 × 4 = 36.
6. Add: 3 + 36 = 39. The 10th term is 39.

Finding the common difference: Subtract any term from the one right after it. In 3, 7, 11 the difference is 7 - 3 = 4. If the numbers go down, d is negative.

• The nth term is a_n = a_1 + (n - 1)d.
• The common difference d is the fixed step between terms.
• It uses (n - 1) because reaching the nth term takes one fewer step than its position.
• A downward sequence just has a negative d.

Why the formula works

1. Start at the first term, a_1.
2. To get to the 2nd term, add the step once: a_1 + d.
3. To get to the 3rd term, add the step twice: a_1 + 2d. To the 4th, three times: a_1 + 3d.
4. The pattern is clear: to reach the nth term you add the step (n - 1) times, which is a_1 + (n - 1)d.

Practice questions

Question 1

Find the 5th term of 2, 5, 8, 11, ...

1. Identify the parts: first term a_1 = 2, common difference d = 3, and n = 5.
2. Write the formula: a_n = a_1 + (n - 1)d.
3. Put in the numbers: a_5 = 2 + (5 - 1) × 3.
4. Work the bracket, then multiply: (5 - 1) × 3 = 4 × 3 = 12.
5. Add: 2 + 12 = 14. The 5th term is 14.

Answer: 14

Question 2

Find the 8th term of 10, 15, 20, ...

1. Identify the parts: first term a_1 = 10, common difference d = 5, and n = 8.
2. Write the formula: a_n = a_1 + (n - 1)d.
3. Put in the numbers: a_8 = 10 + (8 - 1) × 5.
4. Work the bracket, then multiply: 7 × 5 = 35.
5. Add: 10 + 35 = 45. The 8th term is 45.

Answer: 45

Question 3

What is the common difference of 20, 17, 14, 11, ...?

1. The common difference is the fixed step from one term to the next.
2. Subtract any term from the one right after it: 17 - 20 = -3.
3. Check with the next pair: 14 - 17 = -3, the same step.
4. The numbers go down, so the common difference is -3.

Answer: -3

Question 4

Find the 6th term of 1, 4, 7, 10, ...

1. Identify the parts: first term a_1 = 1, common difference d = 3, and n = 6.
2. Write the formula: a_n = a_1 + (n - 1)d.
3. Put in the numbers: a_6 = 1 + (6 - 1) × 3.
4. Work the bracket, then multiply: 5 × 3 = 15.
5. Add: 1 + 15 = 16. The 6th term is 16.

Answer: 16

Question 5

The first term is 100 and the common difference is -10. Find the 4th term.

1. Identify the parts: first term a_1 = 100, common difference d = -10, and n = 4.
2. Write the formula: a_n = a_1 + (n - 1)d.
3. Put in the numbers: a_4 = 100 + (4 - 1) × (-10).
4. Work the bracket, then multiply: 3 × (-10) = -30.
5. Add: 100 + (-30) = 70. The 4th term is 70.

Answer: 70