Let X 1 , X 2 , X 3 . . . . are in arithmetic progression with a common difference equal to d which is a two digit natural number. y 1 , y 2 , y 3 . . . . are in geometric progression with common ratio equal to 16 . Arithmetic mean of X 1 , X 2 . . . . X n is equal to the arithmetic mean of y 1 , y 2 . . . . y n which is equal to 5 . If the arithmetic mean of X 6 , X 7 . . . . X n + 5 is equal to the arithmetic mean

Step-by-step solution

Mean X 1 , X 2 … X n = n 2 2 X 1 + n - 1 d n = 5 Mean of y 1 , y 2 . . . . y n = y 1 16 n - 1 15 n = 5 2 X 1 + n - 1 d = 10 … 1 y 1 16 n - 1 = 75 n … 2 Mean of X 6 , X 7 . . . . X n + 5 = Mean of y P + 1 , y P + 2 , y P + n n 2 2 X 6 + n - 1 d n = y P + 1 16 n - 1 15 n = y 1 16 P 16 n - 1 15 n X 6 + n - 1 2 d = 16 P 75 n 15 n = 5 × 16 P X 1 + 5 d + n - 1 2 d = 5 × 16 P 5 - n - 1 2 d + 5 d + n - 1 2 d = 5 × 16 P ⇒ d = 16 P - 1 ∵ d is 2 digit natural number ⇒ P = 1 , d = 15