Sequences and Series — AP, GP, HP & Convergence

A sequence is an ordered list of numbers. A series is what you get when you add terms from that list. In this topic, AP means arithmetic progression, GP means geometric progression, HP means harmonic progression, and convergence asks whether terms or partial sums approach a finite value.

If you need the short version: AP has a constant difference, GP has a constant ratio, and HP is a sequence whose reciprocals form an AP. For infinite geometric series, the sum exists only when $|r| < 1$.

Sequence vs. series: know which question you are answering

If you write the list

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

you have a sequence. If you write the sum

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

you have a series.

That difference tells you which tool to use. "Find the $n$th term" is a sequence question. "Find the sum of the first $n$ terms" is a series question.

AP, GP, and HP: how to identify each pattern

Arithmetic Progression (AP)

An AP changes by the same amount each step. If the first term is $a$ and the common difference is $d$, then

$$a_n = a + (n-1)d$$

and the sum of the first $n$ terms is

$$S_n = \frac{n}{2}\left[2a + (n-1)d\right]$$

or equivalently

$$S_n = \frac{n}{2}(a + a_n)$$

Example: $4, 7, 10, 13, \dots$ is an AP because each term increases by $3$.

Geometric Progression (GP)

A GP changes by the same factor each step. If the first term is $a$ and the common ratio is $r$, then

$$a_n = ar^{n-1}$$

and for $r \ne 1$,

$$S_n = \frac{a(1-r^n)}{1-r}$$

For an infinite geometric series, the sum exists only when $|r| < 1$. In that case,

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

Example: $3, 6, 12, 24, \dots$ is a GP because each term is multiplied by $2$.

Harmonic Progression (HP)

An HP is defined through reciprocals. A nonzero sequence $a_1, a_2, a_3, \dots$ is in HP if

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

is an AP.

So if

$$\frac{1}{a_n} = A + (n-1)d$$

with nonzero denominator, then

$$a_n = \frac{1}{A + (n-1)d}$$

Example: $\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \dots$ is an HP because its reciprocals $2, 4, 6, 8, \dots$ form an AP.

HP is mainly a classification idea in school math. Unlike AP and GP, it does not come with one standard introductory sum formula that you use in most basic problems.

Convergence: when an infinite process has a finite limit

A sequence converges if its terms approach a fixed limit.

For example,

$$\frac{1}{n} \to 0 \quad \text{as } n \to \infty$$

so the sequence $\left(\frac{1}{n}\right)$ converges to $0$.

A series converges if its partial sums approach a fixed limit. If

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

and the numbers $S_n$ approach some finite value $S$, then the infinite series converges to $S$.

This is the point many students miss: a convergent sequence does not automatically produce a convergent series. The terms going to $0$ are necessary for series convergence, but that condition alone is not enough.

For instance, the harmonic sequence

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

does converge to $0$ as a sequence of terms, but 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 the infinite geometric series

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

This comes from the GP

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

Here the first term is $a = 6$ and the common ratio is

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

Because $|r| = \frac{1}{2} < 1$, the infinite series converges. Its sum is

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

The key step is checking the condition before using the formula. If $|r| < 1$, an infinite geometric series converges. If $|r| \ge 1$, it does not converge to a finite sum.

Common mistakes with sequences, series, and convergence

Mixing up a term and a sum

The term $a_5$ and the sum $S_5$ are not the same kind of answer. One is a term in a list. The other is a total.

Using a difference test on a GP

If the pattern is multiply by $2$, it is geometric even if the numbers are increasing steadily. Constant difference and constant ratio are different tests.

Forgetting the convergence condition for an infinite GP

The formula

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

works 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 is not just a sequence of fractions. Its reciprocals must form an AP.

Where AP, GP, HP, and convergence are used

AP models steady additive change, such as saving the same amount each month. GP models repeated multiplication, such as compound growth or repeated decay. HP appears in school algebra and in problems where reciprocal relationships are the natural pattern.

Convergence matters whenever the process is infinite or very long. It shows up in infinite series, approximation methods, finance, and later topics such as power series and calculus.

Try a similar problem

Take the GP

$$8,\ 4,\ 2,\ 1,\dots$$

Find the common ratio, then decide whether the infinite series $8 + 4 + 2 + 1 + \dots$ converges. After that, compare it with the AP $8, 4, 0, -4, \dots$ to see how quickly the "difference vs. ratio" test separates the two patterns.

If you want a next step, try your own version with a different first term and ratio, and check the convergence condition before you compute any infinite sum.