Question

Which of the following are AP's? If they form an AP, find the common difference d and write three more terms.
(i) −1.2,−3.2,−5.2,−7.2...
(ii)0,-4,−8,−12…
(iii)−$\frac{1}{2}$,−$\frac{1}{2}$,−$\frac{1}{2}$,−$\frac{1}{2}$…
(iv)$\sqrt{2}$,$\sqrt{8}$, $\sqrt{18}$,$\sqrt{32}$…
(v)12,32,52,72…

Answer 1

(i)

−1.2, −3.2, −5.2, −7.2....

It is an AP because difference between consecutive terms is same.

−3.2− (−1.2) =−5.2− (−3.2) =−7.2− (−5.2) = −2

$\Rightarrow$ Common difference (d) = −2

Fifth term = −7.2−2=−9.2
Sixth term = −9.2−2=−11.2
Seventh term = −11.2−2=−13.2

Therefore, next three terms are −9.2, −11.2 and −13.2

(ii)

0, −4, −8, −12...

It is an AP because difference between consecutive terms is constant.

−4−0=−8− (−4) =−12− (−8) =−4

$\Rightarrow$ Common difference (d) = −4

Fifth term = −12−4=−16

Sixth term = −16−4=−20

Seventh term = −20−4=−24

Therefore, next three terms are −16, −20 and −24

(iii)

−12, −12, −12, −12...

It is an AP because difference between consecutive terms is constant.

−12− (−12) =−12 − (−12) =−12 − (−12) =0

$\Rightarrow$ Common difference (d) = 0

Fifth term = −12+0=−12

Sixth term = −12+0=−12

Seventh term = −12+0=−12

Therefore, next three terms are −12, −12 and −12

(iv)

$\sqrt{2}$, $\sqrt{8}$, $\sqrt{18}$, $\sqrt{32}$….

$\sqrt{8}$=2$\sqrt{2}$

$\sqrt{18}$=3$\sqrt{2}$

$\sqrt{32}$=4$\sqrt{2}$

Therefore, sequence is like $\sqrt{2}$, 2$\sqrt{2}$, 3$\sqrt{2}$, 4$\sqrt{2}$…

It is an AP because difference between consecutive terms is constant.

2$\sqrt{2}$ - $\sqrt{2}$ = 3$\sqrt{2}$ - 2$\sqrt{2}$ = 4$\sqrt{2}$ - 3$\sqrt{2}$ = $\sqrt{2}$

$\Rightarrow$ Common difference (d) = $\sqrt{2}$

Fifth term = 4$\sqrt{2}$ + $\sqrt{2}$ = 5$\sqrt{2}$

Sixth term = 5$\sqrt{2}$ + $\sqrt{2}$ = 6$\sqrt{2}$

Seventh term = 6$\sqrt{2}$ + $\sqrt{2}$ = 7$\sqrt{2}$

Therefore, next three terms are 5$\sqrt{2}$, 6$\sqrt{2}$ and 7$\sqrt{2}$

(v)

$1^2$,$3^2$,$5^2$, $7^2$...

It is not an AP because difference between consecutive terms is not constant.

Example: $3^2$ − $1^2$ $\ne$ $5^2$ − $3^2$