Business Mathematics and Basic Statistics · Ch 7 — Arithmetic and Geometric Progressions

Sum of the First n Terms of an AP and a GP

The sum of the first $n$ terms of a progression, denoted $S_n$ , also has a direct formula for both AP and GP — useful whenever a running total (total savings over several months, total production over several years) is needed, rather than a single term.

Note

Sum-to- $n$ -Terms Formulas

Arithmetic Progression: $S_n = \frac{n}{2}\big[2a+(n-1)d\big] = \frac{n}{2}(a+l)$ where $l=T_n$ is the last (i.e. $n$ -th) term — the second form is convenient whenever the last term is already known.
Geometric Progression (for $r \neq 1$ ): $S_n = \frac{a(r^n-1)}{r-1} \quad\text{(convenient when } r>1\text{)}, \qquad S_n=\frac{a(1-r^n)}{1-r} \quad\text{(the same formula, convenient when } 0<r<1\text{)}$ (If $r=1$ , every term equals $a$ , so simply $S_n = na$ .)

Where the AP formula comes from: pairing the first and last term, the second and second-last term, and so on, each pair sums to the SAME total $a+l$ — there are $n/2$ such pairs (this pairing idea, credited to the young Gauss, is why the formula works for any AP, not just special cases). …

Definition 6Sum to n terms, Sn

$S_n = T_1+T_2+\cdots+T_n$ , the total of the first $n$ terms of a progression; has a direct closed-form formula for …

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