Question

Show that each of the progressions given below is an AP. Find the first term, common difference and next term of each.
(i) 9,15,21,27,.....
(ii) 11,6,1,-4,....
(iii) $-1,\frac{-5}{6},\frac{-2}{3},\frac{-1}{2},$...
(iv) $\sqrt{2},\sqrt{8},\sqrt{18},\sqrt{32}$,...
(v) $\sqrt{20},\sqrt{45},\sqrt{80},\sqrt{125}$,....

Answer 1

(i) 9, 15, 21,27…..

Clearly (15 – 9) = (21 – 15) = (27-21) = 6 which is constant

Thus, each term differs from its preceding term by 6.

So, the given progression is an AP.

Next term of the AP = 27 + 6 = 33

Its first term = 9, common difference = 6 and the next term is 33.

(ii) 11, 6, 1, -4….

Clearly (6-11) = (1 – 6) = (-4 – 1) = – 5 which is constant.

Thus, each term differs from its preceding term by -5.

So the given progression is an AP.

Next term of the AP = -4 +(-5) = -9

Its first term = 11 , common difference = – 5 and the next term is -9

(iii) −1,−5/6,−2/3,−1/2,……

Clearly −5/6 – (−1)= -2/3 - (-5/6) = −1/2 − (-2/3) = 1/6

Thus, each term differs from its preceding term by 1/6. So, the given progression is an AP.

First term = -1

Common difference = 1/6

Next term of the AP = −1/2+1/6 = −2/6 = −1/3

(iv) √2,√8,√18,√32......

The given progression √2,√8,√18,√32,…….

This sequence can be re-written as √2, 2√2, 3√2, 4√2…….

Thus, each term differs from its preceding term by √2. So, the given progression is an AP.

First term = √2

Common difference = √2

Next term of an AP = 4√2+√2=5√2

(v) √20,√45,√80,√125…….

This sequence can be re-written as 2√5,3√5,4√5,5√5…….

Clearly d = 3√5-2√5=√5

Thus, each term differs from its preceding term by √5. So, the given progression is an AP.

First term = √20

Common difference =√5

Next term of the AP = 5√5+√5=6√5=√180