How to Find the number of terms in the arithmetic progression

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Finding the number of terms in an arithmetic progression sounds complicated at first, but is actually quite simple. All you need to do is put the given values into the formula t _n = a + (n – 1) d and do the math to find n . Where: n is the number of terms, t _n is the last term in the sequence, a is the first term in the sequence, and d is the difference.

Table of Contents

Steps

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Determine the first, second, and last terms in the arithmetic progression. Usually to solve a problem like this, you will be given at least the first 3 terms and the last term. ^[1]

• For example, the following arithmetic progression: 107, 101, 95…-61. We have: the first term is 107, the second term is 101 and the last term is -61. This is the information you need to solve the problem.

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Subtract the first term from the second term to find the difference. In the example, the first term is 107 and the second term is 101. So the difference will be 107 – 101 = -6. ^[2]

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Apply the formula t _n = a + (n – 1) d to find n . Substitute the last term ( t _n ), the first term ( a ) and the difference ( d ) into the formula. Then do the math until you find the value n . ^[3]

• In the above example, after replacing the values into the formula, we have: -61 = 107 + (n – 1) (-6). Subtract 107 from both sides, leaving -168 = (n – 1) (-6). Then, keep dividing both sides by -6, we get 28 = n – 1. Finally, add 1 to both sides, the answer will be n= 29.

Advice

• The difference of the first term and the last term is always divisible by the common difference.

Warning

• Avoid confusing the difference between the first and last terms with the difference.

Finding the number of terms in an arithmetic progression sounds complicated at first, but is actually quite simple. All you need to do is put the given values into the formula t _n = a + (n – 1) d and do the math to find n . Where: n is the number of terms, t _n is the last term in the sequence, a is the first term in the sequence, and d is the difference.

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