In an arithmetic sequence where a6=13 and a10=25, what is the explicit formula for the general term an?

Answer 1

an = ao + (n-1)d
where
an = the nth term
ao = the first term
n = number of terms
d = difference between two consecutive terms

Substituting the given,
(i) a6 = 13 (here, n=6, an=13)
13 = ao + (6-1)d
13 = ao + 5d

(ii) a10 = 25 (here, n=10, an=25)
25 = ao + (10-1)d
25 = ao + 9d

Now we have two linear equations. We can do elimination method to get d:
13 = ao + 5d
-(25 = ao + 9d)
--------------------------
-12 = -4d
d = 12/4
d = 3

Solving for ao,
13 = ao + 5(3)
13 = ao + 15
ao = 13 - 15
ao = -2

Therefore, the general expression for an:
an = -2 + 3*(n-1)

Hope this helps~ `u`

Answer 2

3n-5