How do you find an explicit formula?

Question

I'm totally lost here. I have an example problem here, so if you could - please help me out!

Example:

"Consider the arithmetic sequence 5, 8, 11, 14, . . .

a. Find an explicit formula for the "n"th term of the sequence
b. Describe how to find the formula in part a"

Answer 1

There are other ways... but all ways must be very difficult... depending of the sequence.

The sequence is a recursion relation that can be written
f(1) =5 and f(n) = f(n-1) + 3 for n>1
this is not a explicit formula... it is a recursion formula.

A way to find the explicit formula is expanding it till you find(if you can) a repeat proceeding.

f(n) = f(n-1) +1x3
= f(n-2) + 2x3
= f(n-3) + 3x3
= f(n-4) + 4x3
=f(n-5) + 5x3

... and then

= f(n-k) +kx3 (general term)

We will reach to the last term when n-k=1 and then k=n-1

Replacing this in the general term, we find f(n) = f(1) + (n-1)x3

and finally: f(n) = 5 + (n-1)x3 or f(n) = 3n+2... this is the explicit formula.

But, attention: you must find a proof of this formula using the induction method.

The other way is more complex.

Answer 2

5+3=8
8+3=11
11+3=14

Formula:
N=1,2,3,4,5...
2+3N