Question

Write the first term, common difference and $n^{th}$ term of A.P.

	A.P.	First term	Common difference	$n^{th}$ term
(i)	-2,2,6,10, ......
(ii)	$\frac{1}{3}$,$\frac{5}{3}$,$\frac{9}{3}$,$\frac{13}{3}$, .....

Answer 1

An A.P has same difference between

any two comsecutive terms called the

common difference of the A.P

Formula for $n^{th}$ term, $a_n$ of an A.P is

gicen by, $a_n=a+(n-1)d$ where a is

the first term and d is the common

difference. Hence we would get

$\left(i\right)-2,2,6,10$

$a=-2;d=2-(-2)=4$

$a_n=-2+(n-1)4=-2+4n-4=4_n-6$

$\left(ii\right)\frac{1}{3},\frac{5}{3},\frac{9}{3},\frac{13}{3},....$

$a=\frac{1}{3};d=\frac{5}{3}-\frac{1}{3}=\frac{4}{3}$

$a_n=\frac{1}{3}+(n-1)\frac{4}{3}=\frac{1}{3}+\frac{4n}{3}-\frac{4}{3}=\frac{4n}{3}-1$

Derivation of $a_n$

We can write terms of an A.P as :

$a,a+d,a+2d,a+3d,...$

(From the property mentioned at the begining)

Therefore, we can see a pattern to write

any $n^{th}$ term as $a_n=a+(n-1)d$.