Question down there to big for up here..?

Question

We define the sequence a recursively by setting
a_1 = 15 and a_(i+1) =a_i + 4 for all i>=1.
This sequence is arithmetic. Give the explicit description of the sequence.

Answer 1

A sequence is said to be arithmetic if the difference d between two adjacent terms a[i+1] and a[i] is constant.

The formula for the i-th term can be derived inductivly:

a[i+1] - a[i] = d => a[i+1] = a[i] + d

a[2] = a[1] + d
a[3] = a[2] + d = (a[1] + d) + d = a[1] + 2*d
a[4] = a[3] + d = (a[1] + 2*d) + d = a[1] + 3*d
....
We notice the regularity:
a[i] = a[1] + (i - 1)*d, i >= 1
This can be proven by mathemctical induction:
a[1] = a[1] + (1-1)*d = a[1] (true!)
a[i+1] = a[i] + d = (a[1] + (i-1)*d) + d = a[1] + i*d (true for i+1)

In our case, a[1] = 15 and d = 4, so the general term is:
a[1] = 15 + 4*(i - 1) = 4*i + 11

Answer 2

Its linear
a_i = 4(i-1) + 15