The 9th term is 9+8×6=57
Sum=9+15+21+27+...+51+57.
This can be written 9+(9+6)+(9+12)+(9+18)+...+(9+42)+(9+48). We have nine 9s.
And this is simplified to 9×9+6+12+18+...+42+48=
81+6(1+2+3+...+7+8).
Look at the sum in parentheses. This can be simplified by pairing numbers:
(1+8)+(2+7)+(3+6)+(4+5)=9+9+9+9=36. There are 4 9s because we have 8/2=4 pairs, and each pair sums to 9.
So the sum of the series is 81+6×36=81+216=297.
If you prefer formulas, let a be the first term, d the common difference and n the number of terms. We have ½(n-1) pairs and each pair sums to 1+(n-1)=n.
The sum Sn=an+½dn(n-1). Let's see if this is correct using n=a=9, d=6:
S9=81+½(6×9×8)=81+216=297. So the formula is correct.