Explanation
Let, a and b two given numbers. Let, A 1 , A 2 , A 3 . . . . . . . . . . . . . A n be Arithmetic means between a and b. a , A 1 , A 2 , A 3 . . . . . . . . . . . . . A n , b are in A.P. a + A 1 + A 2 + A 3 + . . . . . . A n + b = ( 2 n + 2 ) ( a + b ) (first term is =a, last term=b, number of term=n+2) A 1 , A 2 , A 3 . . . . . . . . . . . . . A n = 2 ( n + 2 ) ( a + b ) − ( a + b ) = ( a + b ) [ 2 n + 2 − 1 ] = ( a + b ) [ 2 n + 2 − 2 ] = n ( 2 a + b ) SH14-8-2023