Question

If a1, a2, a3,...... are in AP then ap, aq, ar, are in AP if p, q, r are in
(a) AP
(b) GP
(c) HP
(d) none of these​

Answer 1

p, q, r are in AP if a₁ , a₂ , a₃ ...................... are in AP and ap, aq, ar, are in AP

Step-by-step explanation:

a₁ , a₂ , a₃ ...................... are in AP

d is common difference

=> a₂ = a₁ + d

aₙ = a₁ + (n - 1)d

d is common difference

ap = a₁ + (p - 1)d

aq = a₁ + (q - 1)d

ar = a₁ + (r - 1)d

ap, aq, ar, are in AP

=> ap + ar = 2 aq

=> a₁ + (p - 1)d + a₁ + (r - 1)d = 2 ( a₁ + (q - 1)d)

=> pd + rd = 2qd

Dividing by d from both sides

=> p + r = 2q

=> p , q & r are in AP

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