# explanation of formula for sum of sequence

#### ugkwan

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

9,13,17,21,....

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.

#### SpringFan25

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.

$$\displaystyle a_n = 5+4n$$
$$\displaystyle S_n = 0.5n(a_1 + a_n)$$
$$\displaystyle S_n = 0.5n(9 + (5+4n))$$
$$\displaystyle S_n = 0.5n(14 + 4n)$$
$$\displaystyle S_n = n(7 + 2n)$$

• ugkwan

#### ugkwan

That was a excellent explanation!! You would make a good professor someday.