Question

what is the arithmetic mean of 3.8+6.11+9.14+..............

Answer 1

$T_n = 3n * [ 8 + (n-1) 3] \\ \\ T_n = 24 n + 9 n(n-1) = 15 n + 9 n^2 \\ \\ Sum_n = 15 [ n (n+1) ]/2 + 9 [ n (n+1)(2n+1) ] / 6 \\ \\ = 15/2 n^2 + 15/2 n + 3/2 n [ 2n^2 + 3n + 1 ] \\ \\ Arithmetic mean = \frac{1}{n}\ Sum_n = 15 n/2 + 15/2 + 3/2 [ 2n^2 + 3n+1] \\ \\ = 3 n^2 + 12 n + 9$