6.2 Arithmetic Progression (A.P.)

6.2 Arithmetic Progression (A.P.)

A sequence $\{a_n\}$ is an arithmetic progression if $a_n - a_{n-1} = d$ (constant) for all $n > 1$. The constant $d$ is the common difference.

Examples: $1, 3, 5, 7, \ldots$ ($d = 2$); $54, 51, 48, \ldots$ ($d = -3$).

With $n$ terms: $a_1, a_1+d, \ldots, a_1+(n-1)d$.

Notes:

• Three numbers $a, b, c$ are in A.P. iff $2b = a + c$.
• If $a_1$ is unknown: $a_n = a_m + (n-m)d$ for $n \ge m$.
• Even $n$: two middle terms $\bigl(\tfrac{n}{2}\bigr)$th and $\bigl(\tfrac{n}{2}+1\bigr)$th.
• Odd $n$: one middle term $\bigl(\tfrac{n+1}{2}\bigr)$th.