Class 10

Math

All topics

Arithmetic Progressions

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) $2,25 ,3,27 ,...$

(iii) $−1.2,−3.2,−5.2,−7.2,...$

(iv) $−10,−6,−2,2,...$

(v) $3,3+2 ,3+22 ,3+32 $

(vi) $0.2,0.22,0.222,0.2222,....$

(vii) $0,−4,−8,−12,...$

(viii) $−21 ,−21 ,−21 ,−21 ,...$

(ix) $1,3,9,27$

(x) $a,2a,3a,4a,...$

(xi) $a,a_{2},a_{3},a_{4},...$

(xii) $2 ,8 ,18 ,32 ,...$

(xiii) $3 ,6 ,9 ,12 ,...$

(xiv) $1_{2},3_{2},5_{2},7_{2},..$

(xv) $1_{2},5_{2},7_{2},73,..$

$(i)$ It is not in AP, as the difference between consecutive terms is different.

$(ii)$ It is in AP with common difference $d=25 −2=21 $,

$t_{n}=a+(n−1)d$

$a=2$

$t_{5}=2+(5−1)21 $

Next three terms are $4,29 ,5$

Next three terms are $4,29 ,5$

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

Next
three terms are

$a+(5−1)d=−9.2,$

$a+(6−1)d=−11.2,$

$a+(7−1)d=−13.2$

$(iv)$ It is in AP with common difference $d=−6+10=4$, and

$(iv)$ It is in AP with common difference $d=−6+10=4$, and

$a=−10$

Next three
terms are

$a+(5−1)d=6,$

$a+(6−1)d=10,$

$a+(7−1)d=14$

$(v)$ It is in AP with common difference $d=3+2 −3=2 $, and

$(v)$ It is in AP with common difference $d=3+2 −3=2 $, and

$a=3$

Next
three terms are

$a+(5−1)d=3+42 ,$

$a+(6−1)d=3+52 ,$

$a+(7−1)d=3+62 $

$(vi)$ It is not in AP since $0.22−0.2=0.222−0.22$

$(vii)$ It is in AP with common difference $d=−4−0=−4$ and $a=0$,

$(vi)$ It is not in AP since $0.22−0.2=0.222−0.22$

$(vii)$ It is in AP with common difference $d=−4−0=−4$ and $a=0$,

Next three
terms are

$a+(5−1)d=−16,$

$a+(6−1)d=−20,$

$a+(7−1)d=−24$

$(viii)$ It is in AP, with common difference $0$, therefore next three terms will also be same as previous ones, i.e., $−21 $

$(ix)$ It is not in AP since $3−1=9−3$

$(x)$ It is in AP with common difference $d=2a−a=a$ and first term is $a$,

$(viii)$ It is in AP, with common difference $0$, therefore next three terms will also be same as previous ones, i.e., $−21 $

$(ix)$ It is not in AP since $3−1=9−3$

$(x)$ It is in AP with common difference $d=2a−a=a$ and first term is $a$,

Next
three terms are

$a+(5−1)d=5a,$

$a+(6−1)d=6a,$

$a+(7−1)d=7a$

$(xi)$ It is not in AP, as the difference is not constant.

$(xii)$ It is in AP with common difference $d=2 $ and $a=2 $,

$(xi)$ It is not in AP, as the difference is not constant.

$(xii)$ It is in AP with common difference $d=2 $ and $a=2 $,

Next
three terms are

$a+(5−1)d=52 =50 ,$

$a+(6−1)d=72 ,$

$a+(7−1)d=98 $

$(xiii)$ It is not in AP as difference is not constant.

$(xiii)$ It is not in AP as difference is not constant.

$(xiv)$ It is not in AP as difference is not constant.

$(xv)$ It is in AP with common difference $d=5_{2}−1=24$ and $a=1$,

$(xv)$ It is in AP with common difference $d=5_{2}−1=24$ and $a=1$,

Next three
terms are

$a+(5−1)d=97,$

$a+(6−1)d=121,$

$a+(7−1)d=145$