explanation of formula for sum of sequence

Feb 2010
The question ask to find a formula for the sum of the first n terms of the sequence


So the answer is Sn= n(2n + 7)

Can someone explain why would this arithmetic sequence of sum not look like the usual formula, that is Sn= [n(a1 + an)] / 2

I think Im missing a key point in understanding whats going on here.

Thanks in advance
May 2010

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.

\(\displaystyle a_n = 5+4n\)
\(\displaystyle S_n = 0.5n(a_1 + a_n)\)
\(\displaystyle S_n = 0.5n(9 + (5+4n))\)
\(\displaystyle S_n = 0.5n(14 + 4n)\)
\(\displaystyle S_n = n(7 + 2n)\)
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Feb 2010
That was a excellent explanation!! You would make a good professor someday.