Which of the following are APs? If they form an A.P. find the common difference d and write three more terms.
(i) 0, - 4, - 8, - 12 …
(ii) -1/2, -1/2, -1/2, -1/2 ….
(iii) 1, 3, 9, 27 …
(i) Given, 3, 3+√2, 3+2√2, 3+3√2
a2 – a1 = 3+√2-3 = √2
a3 – a2 = (3+2√2)-(3+√2) = √2
a4 – a3 = (3+3√2) – (3+2√2) = √2
Since, an+1 – an or the common difference is same every time.
Therefore, d = √2 and the given series forms a A.P.
Hence, next three terms are;
a5 = (3+√2) +√2 = 3+4√2
a6 = (3+4√2)+√2 = 3+5√2
a7 = (3+5√2)+√2 = 3+6√2
(ii) 0.2, 0.22, 0.222, 0.2222 ….
a2 – a1 = 0.22-0.2 = 0.02
a3 – a2 = 0.222-0.22 = 0.002
a4 – a3 = 0.2222-0.222 = 0.0002
Since, an+1 – an or the common difference is not same every time.
Therefore, and the given series doesn’t forms a A.P.
(iii) 0, -4, -8, -12 …
a2 – a1 = (-4)-0 = -4
a3 – a2 = (-8)-(-4) = -4
a4 – a3 = (-12)-(-8) = -4
Since, an+1 – an or the common difference is same every time.
Therefore, d = -4 and the given series forms a A.P.
Hence, next three terms are
a5 = -12-4 = -16
a6 = -16-4 = -20
a7 = -20-4 = -24