Series

A series adds the terms of a sequence; arithmetic and geometric structures provide efficient finite-sum formulas.

A sequence tracks terms; a series tracks their accumulating partial sums. Indexed terms appear along a horizontal progression above partial-sum markers. Shrinking term arrows and a target band distinguish terms approaching zero from partial sums actually converging.

A sequence lists terms. A series adds them.

From sequence to sum

If a sequence is $t_1,t_2,t_3,\ldots$, its first $n$ terms form the finite series

S_n=t_1+t_2+\cdots+t_n.

The symbol $S_n$ names a total, while $t_n$ names one term. Keeping them distinct is essential in financial, pattern, and modelling questions.

Arithmetic series formula

For an arithmetic sequence with first term $a$, last term $t_n$, and $n$ terms,

S_n=\frac n2(a+t_n).

Since $t_n=a+(n-1)d$, an equivalent formula is

S_n=\frac n2\bigl(2a+(n-1)d\bigr).

Use whichever form matches the available information.

Why the arithmetic formula works

Write an arithmetic series forward and backward. Each vertical pair has the same sum $a+t_n$, and there are $n$ pairs across the two copies. Therefore $2S_n=n(a+t_n)$, giving the formula after division by $2$.

This pairing argument explains the structure rather than asking us to memorize it blindly.

Finite geometric series formula

For a geometric sequence with first term $a$ and common ratio $r\ne1$,

S_n=a\frac{1-r^n}{1-r}.

An equivalent form is $a(r^n-1)/(r-1)$. Choose one form and substitute signs carefully. If $r=1$, every term equals $a$, so $S_n=na$.

Why the geometric formula works

Start with

S_n=a+ar+ar^2+\cdots+ar^{n-1}.

Multiply by $r$ and subtract the original sum. Most terms cancel:

$$ rS_n-S_n=ar^n-a. $$

Then $(r-1)S_n=a(r^n-1)$, which rearranges to the finite formula.

Common mistakes

Using a term formula for a total. Identify $t_n$ versus $S_n$.

Using the wrong number of terms. From index $p$ to $q$ inclusive there are $q-p+1$ terms.

Applying the arithmetic formula to a geometric pattern. Test difference and ratio first.

Dropping parentheses around $1-r^n$. Evaluate the exponent before subtraction.

Using $S_\infty$ when $|r|\ge1$. The convergence condition is required.