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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, 2a, 3a, 4a

(xi) a, a2, a3, a4
(xii)

(xiii)

(xiv) 12, 32, 52, 72

(xv) 12, 52, 72, 73 …

Answer:

(i) 2, 4, 8, 16 …

It can be observed that

a2a1 = 4 − 2 = 2

a3a2 = 8 − 4 = 4

a4a3 = 16 − 8 = 8

i.e., ak+1ak is not the same every time. Therefore, the given numbers are not forming an A.P.

(ii)

It can be observed that

i.e., ak+1ak 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

a2a1 = (−3.2) − (−1.2) = −2

a3a2 = (−5.2) − (−3.2) = −2

a4a3 = (−7.2) − (−5.2) = −2

i.e., ak+1ak is same every time. Therefore, d = −2

The given numbers are in A.P.

Three more terms are

a5 = − 7.2 − 2 = −9.2

a6 = − 9.2 − 2 = −11.2

a7 = − 11.2 − 2 = −13.2

(iv) −10, −6, −2, 2 …

It can be observed that

a2a1 = (−6) − (−10) = 4

a3a2 = (−2) − (−6) = 4

a4a3 = (2) − (−2) = 4

i.e., ak+1 ak is same every time. Therefore, d = 4

The given numbers are in A.P.

Three more terms are

a5 = 2 + 4 = 6

a6 = 6 + 4 = 10

a7 = 10 + 4 = 14

(v)

It can be observed that

i.e., ak+1 − ak 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

a2a1 = 0.22 − 0.2 = 0.02

a3a2 = 0.222 − 0.22 = 0.002

a4a3 = 0.2222 − 0....

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