Arithmetic Progressions

5.1

Arithmetic Progressions

$The image shows a diagram explaining two formulas related to Arithmetic Progression. On the left side, there is a blue circle with the text ‘Arithmetic Progression’ inside it and the Pandai logo at the bottom. On the right side, there are two rectangular boxes connected to the circle with red lines. The top box contains the formula for the n-th term of an Arithmetic Progression: $ a_n = a + (n - 1)d $. The bottom box contains the formula for the sum of the first n terms: $ S_n = \frac{n}{2} [2a + (n - 1)d] $.$

Definition

Arithmetic Progression (AP) is a sequence of numbers such that each term after the first is obtained by adding the previous one with a constant called common difference, $d$.

General Form

The $n$-th term of an arithmetic progression is given by:

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

Where:
$a=$ first term,
$d=$ common difference,
$n=$ term number,
$T_n=n$-th term.

Common Difference, $d$

Formula to find the common difference:

$d=T_n-T_{n-1}$

Example:
For the sequence $3$, $7$, $11$, $15$, the common difference, $d$:
$d=7-3=4$.

Sum of the First $n$ Terms, $S_n$

The sum of the first $n$ terms, $S_n$, is given by:

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

Alternatively, it can be written as:

$S_n=\dfrac{n}{2}(a+T_n)$

Examples

Finding the $n$-th Term

Given $a=5$, $d=3$, find the $10^{\text{th}}$ term.
Solution:
$\begin{aligned} T_{10}&=5+(10-1)\times 3 \\ &=5+27 \\ &=32. \end{aligned}$

Finding the Sum of the First $n$ Terms

Given $a=2$, $d=4$, find the sum of the first $8$ terms.
Solution:
$\begin{aligned} S_8&=\dfrac{8}{2}[2\times2+(8-1)\times4] \\ &=4[4+28] \\ &=4\times32 \\ &=128. \end{aligned}$