The Nth Term Formula

To jump to any position without listing every number, use:

aₙ = a₁ + (n − 1)d

Here a₁ is the first term, d is the common difference, and n is the position you want. Try the sequence 5, 8, 11, 14, where d is 3. The 100th term is 5 + (100 − 1) × 3 = 5 + 297 = 302.

No counting through 99 terms first. The formula lands on the answer in one line — which is exactly what you need when a sequence runs to thousands of terms.

Lines In Disguise

The nth-term formula hides a neat truth: every arithmetic sequence is really a straight line. Plot the terms — position on the bottom, value up the side — and the dots fall perfectly in a row.

The common difference d is the slope, the steepness of that line. The first term sets where the line starts. So an arithmetic sequence is just a line you read at whole-number steps. That is why it never curves: a constant step is a constant slope.