Example 3: Find the sum of the arithmetic progression 2, 5, 8, 11, 14, 17?

Solution: We have the first term a=2 and the common difference d=3 and n=6 the total number of terms of the series.

Now all we have to do is substitute the values in the formula

$ \displaystyle {{S}_{n}}=\frac{n}{2}\left[ {2a+(n-1)\left. d \right]} \right.$

$ \displaystyle =\frac{6}{2}\left[ {2(2)+(6-1)3} \right]$

$ \displaystyle ~=3\left[ {4+5\cdot 3} \right]$

$ \displaystyle ~~~~~~=3\left[ {4+15} \right]$

$\displaystyle ~~=3\cdot 19=57$

So the sum is 57.