Nth term of a sequence: the two formulas you must never forget

The two formulas

Arithmetic sequence (you always add the common difference $d$): $u_n = u_0 + n \cdot d$ or if starting from $u_1$: $u_n = u_1 + (n-1) \cdot d$

Geometric sequence (you always multiply by the common ratio $q$): $u_n = u_0 \cdot q^n$ or if starting from $u_1$: $u_n = u_1 \cdot q^{n-1}$

Anti-off-by-one trick

The classic trap: is it $n$ or $n-1$ in the exponent?

Simple rule: count the number of “steps” between the starting index and the target index.

• From $u_0$ to $u_n$: there are $n$ steps → coefficient $n$.
• From $u_1$ to $u_n$: there are $n-1$ steps → coefficient $n-1$.

Example 1

Arithmetic sequence with $u_0 = 3$ and $d = 5$. What is $u_{10}$?

$u_{10} = 3 + 10 \times 5 = 53$.

Example 2

Geometric sequence with $u_1 = 2$ and $q = 3$. What is $u_5$?

From $u_1$ to $u_5$: 4 steps. $u_5 = 2 \times 3^{4} = 162$.