Given that n number of arithmetic means are inserted between two pairs of numbers a, 2 b and 2 a, b ; where a, b R . If the m^ th means in the two cases are the same, then the ratio a: b is equal to

Step-by-step solution

Let the two numbers be x and y . If n arithmetic means A₁, A₂, ..., A_n are inserted between x and y , the common difference d is given by d = y - x n + 1 . The m^ th arithmetic mean is A_m = x + m d = x + m ( y - x n + 1 ) . For the first pair a and 2b , the m^ th mean is M₁ = a + m ( 2b - a n + 1 ) = a(n + 1) + m(2b - a) n + 1 = an + a + 2bm - am n + 1 = a(n - m + 1) + 2bm n + 1 . For the second pair 2a and b , the m^ th mean is M₂ = 2a + m ( b - 2a n + 1 ) = 2a(n + 1) + m(b - 2a) n + 1 = 2an + 2a + bm - 2am n +