Sequences and Series — AP, GP, HP & Convergence

In one line: a sequence is an ordered list of numbers, a series is the sum of terms from that list, and the type of pattern (AP, GP, or HP) decides which formula you use. Here AP means arithmetic progression, GP means geometric progression, and HP means harmonic progression, while convergence asks whether terms or partial sums approach a finite value. AP has a constant difference, GP has a constant ratio, HP is a sequence whose reciprocals form an AP, and an infinite geometric series sums only when $|r| < 1$.

Sequence vs. series, and the three patterns side by side

The list

$$2,\ 5,\ 8,\ 11,\dots$$

is a sequence; the sum

$$2 + 5 + 8 + 11 + \dots$$

is a series. "Find the $n$th term" is a sequence question; "find the sum of the first $n$ terms" is a series question.

The three progressions compare like this:

Type	Pattern rule	$n$th term	Sum (first $n$ terms)	Example
AP	constant difference $d$	$a_n = a + (n-1)d$	$S_n = \frac{n}{2}[2a + (n-1)d]$	$4, 7, 10, 13, \dots$
GP	constant ratio $r$	$a_n = ar^{n-1}$	$S_n = \frac{a(1-r^n)}{1-r}$ ($r \ne 1$)	$3, 6, 12, 24, \dots$
HP	reciprocals form an AP	$a_n = \frac{1}{A + (n-1)d}$	no single standard intro formula	$\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \dots$

For AP, the sum also has the compact form $S_n = \frac{n}{2}(a + a_n)$. For an infinite GP with $|r| < 1$,

$$S_{\infty} = \frac{a}{1-r}.$$

HP is defined entirely through reciprocals. A nonzero sequence $a_1, a_2, a_3, \dots$ is harmonic when

$$\frac{1}{a_1},\ \frac{1}{a_2},\ \frac{1}{a_3},\dots$$

is an AP, so if $\frac{1}{a_n} = A + (n-1)d$ with nonzero denominator, then $a_n = \frac{1}{A + (n-1)d}$. In the example above, the reciprocals $2, 4, 6, 8, \dots$ form an AP, which is what makes the original sequence harmonic. Unlike AP and GP, HP does not come with one standard introductory sum formula, so in school math it stays mostly a classification idea.

When to use each, including convergence

Use AP reasoning for steady additive change, like saving the same amount each month. Use GP reasoning for repeated multiplication, like compound growth or decay. Use HP when reciprocal relationships are the natural pattern. Bring in convergence whenever the process is infinite or very long.

A sequence converges if its terms approach a fixed limit; for example $\frac{1}{n} \to 0$ as $n \to \infty$, so $\left(\frac{1}{n}\right)$ converges to $0$. A series converges if its partial sums

$$S_n = a_1 + a_2 + \dots + a_n$$

approach a finite value $S$. The point many students miss: a convergent sequence does not force a convergent series. Terms going to $0$ is necessary but not sufficient. The harmonic sequence

$$1,\ \frac{1}{2},\ \frac{1}{3},\ \frac{1}{4},\dots$$

converges to $0$, yet the harmonic series $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots$ does not converge to a finite sum.

Worked example: test a GP and sum the infinite series

Consider

$$6 + 3 + \frac{3}{2} + \frac{3}{4} + \dots,$$

from the GP $6,\ 3,\ \frac{3}{2},\ \frac{3}{4},\dots$. The first term is $a = 6$ and the ratio is

$$r = \frac{3}{6} = \frac{1}{2}.$$

Because $|r| = \frac{1}{2} < 1$, the series converges, with sum

$$S_{\infty} = \frac{a}{1-r} = \frac{6}{1-\frac{1}{2}} = \frac{6}{\frac{1}{2}} = 12.$$

The key step is checking the condition before using the formula: $|r| < 1$ converges, $|r| \ge 1$ does not. Skipping that check is how students "sum" series that have no finite sum at all.

Exam confusion points to watch

Mixing up a term and a sum. $a_5$ is a term in a list; $S_5$ is a total. Different kinds of answer.

Using a difference test on a GP. If the rule is "multiply by $2$," it is geometric even when numbers rise steadily. Constant difference and constant ratio are different tests.

Forgetting the infinite-GP condition. $S_{\infty} = \frac{a}{1-r}$ holds only when $|r| < 1$.

Thinking "terms go to zero" is enough. For series it is only a first check; the harmonic series is the standard counterexample.

Treating HP as "anything with fractions." An HP requires that the reciprocals form an AP.

Practice this yourself

Take the GP $8,\ 4,\ 2,\ 1,\dots$. Find the common ratio, then decide whether $8 + 4 + 2 + 1 + \dots$ converges. Compare it with the AP $8, 4, 0, -4, \dots$ to see how fast the difference-vs-ratio test separates the two. For an extra pass, change the first term and ratio, and always check the convergence condition before computing any infinite sum.

Where these ideas are used

AP models steady additive change, GP models repeated multiplication, and HP appears in school algebra and reciprocal-pattern problems. Convergence shows up in infinite series, approximation methods, finance, and later topics such as power series and calculus.