Question 847871
Let's look at some things. What is the difference between these numbers?

45 -21 = 24

77-45 = 32

117 - 77 = 40...

etc.  It seems as though our difference is increasing by 8 each time.

By knowing that the 2nd difference is 8, we know that the function is of the form 4n^2 + something. In general, an^2 has second difference 2a.

We also realize that the difference between a_0 and a_1 is 16.

So a_0 = 5.

So we have 4n^2 + b*n + 5

This should be enough to figure out what b is.

Take a_2 for instance.

4(2)^2 + b*2 + 5  = 45

16 + 2b + 5 = 45

2b = 24

b = 12

So our entire function a_n  =  4n^2 + 12n + 5

We check our answer:

a(1) =  4 + 12 + 5 = 21

a(2) =  4*4 + 24 + 5 = 45

a(3) =  4*9 + 36 + 5 = 77

a(4) =  4*16 + 48 + 5 = 117

a(5) =  4*25 + 60 + 5 =  165

Just for fun a(10) = 4*100 + 12*10 + 5 =  525.  Simple.