How to Find Which Term of an AP Is a Given Number

How do you find the term number?

For an arithmetic progression with first term a and nonzero common difference d, solve n = (T − a)/d + 1. The target T belongs to the AP only when n is a positive integer. The formula identifies a position, so a fractional answer cannot be rounded to a nearby term.

Worked example: is 83 in the sequence?

Consider 7, 11, 15, 19, … . Here a = 7 and d = 4. To locate 83, write 83 = 7 + (n − 1)4. Subtracting 7 gives 76 = 4(n − 1), so n − 1 = 19 and n = 20. Therefore 83 is the twentieth term. Check: 7 + 19 × 4 = 83.

Now try 85. The same calculation gives n = 78/4 + 1 = 20.5. This does not mean the twenty-first term: 85 is not a term of this AP. The neighbouring terms are 83 and 87.

What if the sequence decreases?

For 30, 27, 24, …, the difference is −3. To find the position of 6, solve n = (6 − 30)/(−3) + 1 = 9. Negative d is valid. The important check is that n remains a positive integer. If a question limits the AP to its first eight terms, 6 is outside that finite list even though it belongs to the continuing progression.

If d = 0, every term equals a. Then T = a occurs at every allowed position, while any other target never occurs. Do not divide by zero.

Practice and checks

1. Which term of 5, 8, 11, … is 50? 16th, because 5 + 15 × 3 = 50.
2. Does 52 occur in that AP? No: n = 50/3 is not an integer.
3. Which term of 18, 16, 14, … is 0? 10th, because 18 + 9 × (−2) = 0.

Write a, d and T before rearranging. This keeps the value of a term separate from its position.

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