# Arithmetic Progressions

#### Question 4:

Which of the following
are APs? If they form an A.P. find the common difference *d* and
write three more terms.

(i) 2, 4, 8, 16 …

(ii)

(iii) − 1.2, − 3.2, − 5.2, − 7.2 …

(iv) − 10, − 6, − 2, 2 …

(v)

(vi) 0.2, 0.22, 0.222, 0.2222 ….

(vii) 0, − 4, − 8, − 12 …

(viii)

(ix) 1, 3, 9, 27 …

(x) *a*, 2*a*,
3*a*, 4*a* …

(xi) *a*, *a*^{2},
*a*^{3}, *a*^{4} …

(xii)

(xiii)

(xiv) 1^{2},
3^{2}, 5^{2}, 7^{2} …

(xv) 1^{2}, 5^{2},
7^{2}, 73 …

#### Answer:

(i) 2, 4, 8, 16 …

It can be observed that

*a*_{2}
− *a*_{1} = 4 − 2 = 2

*a*_{3}
− *a*_{2}^{ }= 8 − 4 = 4

*a*_{4}
− *a*_{3} = 16 − 8 = 8

i.e.,
*a*_{k}_{+1}− *a*_{k}
is not the same every time. Therefore, the given numbers are not
forming an A.P.

(ii)

It can be observed that

i.e.,
*a*_{k}_{+1}− *a*_{k}
is same every time.

Therefore, and the given numbers are in A.P.

Three more terms are

(iii) −1.2, −3.2, −5.2, −7.2 …

It can be observed that

*a*_{2}
− *a*_{1} = (−3.2) − (−1.2) = −2

*a*_{3}
− *a*_{2 }= (−5.2) − (−3.2) = −2

*a*_{4}
− *a*_{3} = (−7.2) − (−5.2) = −2

i.e.,
*a*_{k}_{+1}− *a*_{k}
is same every time. Therefore, *d* = −2

The given numbers are in A.P.

Three more terms are

*a*_{5}
= − 7.2 − 2 = −9.2

*a*_{6}
= − 9.2 − 2 = −11.2

*a*_{7}
= − 11.2 − 2 = −13.2

(iv) −10, −6, −2, 2 …

It can be observed that

*a*_{2}
− *a*_{1} = (−6) − (−10) = 4

*a*_{3}
− *a*_{2 }= (−2) − (−6) = 4

*a*_{4}
− *a*_{3} = (2) − (−2) = 4

i.e.,
*a*_{k}_{+1 }− *a*_{k}
is same every time. Therefore, *d* = 4

The given numbers are in A.P.

Three more terms are

*a*_{5}
= 2 + 4 = 6

*a*_{6}
= 6 + 4 = 10

*a*_{7}
= 10 + 4 = 14

(v)

It can be observed that

i.e.,
*a*_{k}_{+1 −} *a*_{k}
is same every time. Therefore,

The given numbers are in A.P.

Three more terms are

(vi) 0.2, 0.22, 0.222, 0.2222 ….

It can be observed that

*a*_{2}
− *a*_{1} = 0.22 − 0.2 = 0.02

*a*_{3}
− *a*_{2 }= 0.222 − 0.22 = 0.002

*a*_{4}
− *a*_{3} = 0.2222 − 0.22…

To view the solution to this question please