General term of an arithmetic sequence

The explicit formula

Calculating u_100 by adding the common ratio 100 times would take a long time. We use the formula for the general term: u_n = u_0 + n * r

More generally, if we know a term $u_p$: $u_n = u_p + (n - p) \cdot r$

Example

Let $u_0 = 5$ be the first term of an arithmetic sequence with common difference $r = 4$. u_n = 5 + n * 4 = 5 + 4n

To find u_20: u_20 = 5 + 4*20 = 5 + 80 = 85.

Finding the common difference and the first term

If two terms are known, we can find r. Example: u_3 = 11 and u_7 = 23. r = (u_7 - u_3) / (7 – 3) = (23 – 11) / 4 = 3 Then u_0 = u_3 – 3r = 11 – 9 = 2.

n	Formula	Value
0	2 + 0 × 3	2
3	2 + 3 × 3	11
7	2 + 7 × 3	23

Pitfalls to avoid

• Be careful with the starting index: some sequences start at u_0, others at u_1. The formula must be adapted accordingly: u_n = u_1 + (n-1)*r.
• Do not reverse the numerator and denominator when calculating r.
• Make a clear distinction between n (index) and r (common ratio): a common mistake is to write u_n = n*r without adding the first term.