nth Term of an Arithmetic Progression: Formula Guide

The nth term of an arithmetic progression is the formula that jumps straight to any position in the sequence without listing all the earlier terms. If the first term is a and the common difference is d, then the nth term is T_n = a + (n - 1)d. The (n - 1) appears because moving from the 1st term to the nth term requires exactly (n - 1) jumps of size d.

This formula matters because real sequences are long: if output rises by 4 units daily from 7 on day one, nobody wants to list 200 terms to find day 200's output. The nth-term formula gives it in one line of arithmetic, and it also works backwards to find which position holds a given value.

Key Concepts

• n: the position number of the term. It must be a positive whole number: 1, 2, 3, … .
• a: the first term, which is T_1 itself.
• d: the common difference, equal to any term minus the one before it.
• Why (n - 1): T_1 = a, T_2 = a + d, T_3 = a + 2d. The multiplier of d is always one less than the term number.
• Rearranged forms: n = (T_n - a)/d + 1 locates a known value; d = (T_n - a)/(n - 1) recovers the difference.

Formulas

• nth term: T_n = a + (n - 1)d.
• Finding the position: n = (T_n - a)/d + 1, valid when the value is genuinely a term of the AP.
• Finding the difference: d = (T_n - a)/(n - 1).
• Assumptions: n is a positive integer; d is constant through the sequence.

Worked Example

Find the 50th term of the AP 7, 11, 15, 19, … . Here a = 7 and d = 11 - 7 = 4. Then T_50 = 7 + (50 - 1) times 4 = 7 + 196 = 203. Is 343 a term? n = (343 - 7)/4 + 1 = 84 + 1 = 85, a whole number, so 343 is the 85th term.

Common Misconceptions

• Off-by-one error: writing T_n = a + nd adds one jump too many. Testing n = 1 exposes it: the first term must stay a.
• n can be 0 or a fraction: in the AP formulas, n counts a position, so only positive whole numbers make sense. There is no 2.5th term.
• Negative d needs a different formula: it does not. For 100, 93, 86, … with d = -7, the 5th term is T_5 = 100 + 4 times (-7) = 100 - 28 = 72, which matches the listed sequence.

Real-World Applications

• Planning: predicting production, savings, or seating on a far-future date from a steady daily change.
• Finding missing terms: given the 3rd term is 12 and the 8th term is 32, d = (32 - 12)/5 = 4 and a = 12 - 8 = 4, rebuilding the whole AP.
• Programming: loops that step through arithmetic sequences, such as pagination offsets, use the same formula.

The nth term of an AP, T_n = a + (n - 1)d, reaches any position instantly. Remember that (n - 1) counts the jumps, that n must be a positive whole number, and that the formula also runs backwards to locate a value.