If ( m ) arithmetic means are inserted between 1 and 31 so that the ratio of the (7^ th ) and (( m -1)^ th ) means is (5: 9 ), then find the value of (m ).

Step-by-step solution

Let the means be (x₁, x₂, x_m ) so that (1, x ₁, x ₂, . x _ m , 31 ) is an A.P. of (( m +2) ) terms. Now, (31= T _ m +2 = a +( m +1) d =1+( m +1) d ) ( d = 30 ~m +1 ) Given : ( x ₇ x _ m -1 = 5 9 ) ( aligned & T ₈ ~T _ m = a +7 ~d a +( m -1) d = 5 9 & 9 a +63 ~d =5 a +(5 ~m -5) d & 4.1=(5 ~m -68) 30 ~m +1 & 2 ~m +2=75 ~m -1020 73 ~m =1022 & m = 1022 73 =14 aligned )