Question: If n– arithmetic means are inserted between 1 and 31 such that the 7<sup>th</sup> mean : the (n–1)<s...

Question

If n– arithmetic means are inserted between 1 and 31 such that the 7^th mean : the (n–1)^th mean = 5 : 9, then ‘n’ is equal to

Answer 1

1, a₁, a₂………a_n–1,a_n, 31 are in A.P.

a₁ = 1 + d

a₂ = 1 + 2d : : :

a_n = 1 + nd 31 = 1 + (n +1) d ̃ (n + 1) d = 30

$\frac{7^{th}mean}{(n - 1)^{th}mean}$ = $\frac{5}{9}$ = $\frac{1 + 7d}{1 + (n - 1)d}$

̃ 73 d = 146

d = 2 ̃ (n + 1) 2 = 30

n = 14