Arithmetic Progressions

Number rules the universe. — Pythagoras

5.1 Introduction

An Arithmetic Progression (AP) is a list of numbers in which each term after the first is obtained by adding a fixed number $d$ (the common difference) to the preceding term.

$a,\ a+d,\ a+2d,\ a+3d,\ \ldots$

5.2 nth Term

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

where $a$ is the first term.

5.3 Sum of First n Terms

$S_n=\frac{n}{2}\bigl(2a+(n-1)d\bigr)=\frac{n}{2}(a+\ell)$

where $\ell$ is the last term ($a_n$).

Useful when the number of terms, first term, and common difference are known.

Chapter Summary

• AP: each term differs by constant $d$
• $a_n=a+(n-1)d$
• $S_n=\dfrac{n}{2}(2a+(n-1)d)$

Exercises (NCERT)

• Exercise 5.1 — identify AP, find $d$
• Exercise 5.2 — nth term
• Exercise 5.3 — sum of $n$ terms
• Exercise 5.4 — applications
• Miscellaneous Exercise