Sequence and Series (AP, GP, HP): Practice Questions & Notes

A sequence is an ordered list of numbers following a specific pattern or rule.

Example:
2, 4, 6, 8, 10 is a sequence (in this case, AP)

A series is the sum of terms of a sequence.
If you sum the terms of a sequence, you get a series.

Example:
2 + 4 + 6 + 8 + 10 is a series.

Types of Sequences

We'll focus on three common types:

In an AP, the difference between any two consecutive terms is constant.
This constant is called the common difference (d).

If $a$ is the first term and $d$ is the common difference, then:

\text{AP: } a, a + d, a + 2d, a + 3d, \dots

nth term:
$T_n = a + (n - 1)d$
Sum of first n terms (Sₙ):
$S_n = \frac{n}{2} [2a + (n - 1)d] = \frac{n}{2} (a + l)$
where $l$ is the last term.
Mean of n terms:
$\frac{S_n}{n} = \frac{a + l}{2}$

Mistake	Fix
Using wrong formula for nth term	Double-check if you’re using AP or GP
Assuming AP always starts from 1	Look carefully at the given first term
Forgetting to use last term in sum formula	$S_n = \frac{n}{2}(a + l)$ is easiest when last term is known

Example 1:
Find the 12th term of an AP where $a = 5$ , $d = 3$

T_{12} = 5 + (12 - 1) \cdot 3 = 5 + 33 = 38

Example 2:
Sum of first 50 natural numbers

S_{50} = \frac{50}{2}(1 + 50) = 25 \cdot 51 = 1275

In a GP, the ratio between successive terms is constant.
This is called the common ratio (r).

If $a$ is the first term and $r$ is the ratio:

\text{GP: } a, ar, ar^2, ar^3, \dots

nth term:
$T_n = ar^{n-1}$
Sum of first n terms (Sₙ):
$S_n = a \cdot \frac{r^n - 1}{r - 1} \quad \text{(if } r \neq 1 \text{)}$
Sum of infinite GP (when $|r| < 1$ ):
$S = \frac{a}{1 - r}$

Mistake	Fix
Using infinite sum formula for $r \geq 1$	Only valid if (	r	< 1 )
Mixing AP and GP terms	Always check: is the growth additive (AP) or multiplicative (GP)?
Wrong exponent in nth term	$ar^{n-1}$ , not $ar^n$

Example 1:
Find 6th term of GP: 2, 4, 8, 16, ...

Here, $a = 2$ , $r = 2$

T_6 = 2 \cdot 2^{5} = 2 \cdot 32 = 64

Example 2:
Sum of infinite GP with $a = 3$ , $r = \frac{1}{2}$

S = \frac{3}{1 - \frac{1}{2}} = \frac{3}{0.5} = 6

A harmonic progression is a sequence where the reciprocals of the terms form an AP.

If $a, b, c$ are in HP, then:

\frac{1}{a}, \frac{1}{b}, \frac{1}{c} \text{ are in AP}

There is no direct formula for nth term or sum of HP, but:

nth term of HP:
$H_n = \frac{1}{a + (n-1)d} \quad \text{where } \frac{1}{a} \text{ is the first term of AP}$
Harmonic Mean (HM) between a and b:
$HM = \frac{2ab}{a + b}$

Example 1:
Find the HM of 3 and 6

HM = \frac{2 \cdot 3 \cdot 6}{3 + 6} = \frac{36}{9} = 4

Example 2:
If $1/2, 1/3, 1/4$ are in AP, then corresponding HP is:

2, 3, 4

Feature	AP	GP	HP
Pattern	Additive	Multiplicative	Reciprocal of AP
nth Term	$a + (n - 1)d$	$ar^{n-1}$	$1/[a + (n - 1)d]$
Sum (finite)	Easy	Only closed for $r \neq 1$	No direct formula
Infinite Sum	Not defined	Only if (	r	< 1 )	Not defined

Questions often mix AP/GP in a disguised form — e.g., compound interest disguised as a GP.
Check whether a sequence is additive or multiplicative to determine AP vs GP.
HP questions are rare but frequently asked in technical interviews.
Be alert to wording like:

"Three numbers in AP/GP/HP with given sum/product" — implies solving simultaneous equations.