ukgwan,

if you try your formula \(\displaystyle S_n = 0.5n(a_1 + a_n)\), you'll see that it actually works. The problem is that to do yours, you need to know the values of \(\displaystyle a_1\) and \(\displaystyle a_n\).

The textbook solution \(\displaystyle S_n = n(7 + 2n)\) is quicker, because you just input n and get the answer directly.

Your not missing anything conceptual here, if you work out a formula for \(\displaystyle a_n\), and then subsititute it into your method, you will get the textbook answer.

\(\displaystyle a_n = 5+4n\)

\(\displaystyle S_n = 0.5n(a_1 + a_n)\)

\(\displaystyle S_n = 0.5n(9 + (5+4n))\)

\(\displaystyle S_n = 0.5n(14 + 4n)\)

\(\displaystyle S_n = n(7 + 2n)\)