Question

Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms.
(i) $2,4,8,16,....$
(ii) $2,\frac{5}{2},3,\frac{7}{2},...$
(iii) $-1.2,-3.2,-5.2,-7.2,...$
(iv) $-10,-6,-2,2,...$
(v) $3,3+\sqrt{2},3+2\sqrt{2},3+3\sqrt{2}$
(vi) $0.2,0.22,0.222,0.2222,....$
(vii) $0,-4,-8,-12,...$
(viii) $-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},...$
(ix) $1,3,9,27$
(x) $a,2a,3a,4a,...$
(xi) $a,a^2,a^3,a^4,...$
(xii) $\sqrt{2},\sqrt{8},\sqrt{18},\sqrt{32},...$
(xiii) $\sqrt{3},\sqrt{6},\sqrt{9},\sqrt{12},...$
(xiv) $1^2,3^2,5^2,7^2,..$
(xv) $1^2,5^2,7^2,73,..$

Answer 1

$a,b,c$ are said to be in AP if the common difference between any two consecutive number of the series is same ie $b-a=c-b\Rightarrow 2b=a+c$
$\left(i\right)$ It is not in AP, as the difference between consecutive terms is different.

$\left(ii\right)$ It is in AP with common difference $d=\frac{5}{2}-2=\frac{1}{2}$,

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

$a=2$

$t_5=2+(5-1)\frac{1}{2}$
Next three terms are $4,\frac{9}{2},5$

$\left(iii\right)$ It is in AP with common difference $d=-3.2+1.2=-2$ ,and $a=-1.2$

Next three terms are

$a+(5-1)d=-9.2,$

$a+(6-1)d=-11.2,$

$a+(7-1)d=-13.2$
$\left(iv\right)$ It is in AP with common difference $d=-6+10=4$, and

$a=-10$

Next three terms are

$a+(5-1)d=6,$

$a+(6-1)d=10,$

$a+(7-1)d=14$
$\left(v\right)$ It is in AP with common difference $d=3+\sqrt{2}-3=\sqrt{2}$, and

$a=3$

Next three terms are

$a+(5-1)d=3+4\sqrt{2},$

$a+(6-1)d=3+5\sqrt{2},$

$a+(7-1)d=3+6\sqrt{2}$
$\left(vi\right)$ It is not in AP since $0.22-0.2\ne 0.222-0.22$
$\left(vii\right)$ It is in AP with common difference $d=-4-0=-4$ and $a=0$,

Next three terms are

$a+(5-1)d=-16,$

$a+(6-1)d=-20,$

$a+(7-1)d=-24$
$\left(viii\right)$ It is in AP, with common difference $0$, therefore next three terms will also be same as previous ones, i.e., $-\frac{1}{2}$
$\left(ix\right)$ It is not in AP since $3-1\ne 9-3$
$\left(x\right)$ It is in AP with common difference $d=2a-a=a$ and first term is $a$,

Next three terms are

$a+(5-1)d=5a,$

$a+(6-1)d=6a,$

$a+(7-1)d=7a$
$\left(xi\right)$ It is not in AP, as the difference is not constant.
$\left(xii\right)$ It is in AP with common difference $d=\sqrt{2}$ and $a=\sqrt{2}$,

Next three terms are

$a+(5-1)d=5\sqrt{2}=\sqrt{50},$

$a+(6-1)d=\sqrt{72},$

$a+(7-1)d=\sqrt{98}$
$\left(xiii\right)$ It is not in AP as difference is not constant.

$\left(xiv\right)$ It is not in AP as difference is not constant.
$\left(xv\right)$ It is in AP with common difference $d=5^2-1=24$ and $a=1$,

Next three terms are

$a+(5-1)d=97,$

$a+(6-1)d=121,$

$a+(7-1)d=145$