In an arithmetic sequence a9 = 23 and a14 = 33. What is the 1st term? What is the recursive formula of the sequence?

Answer 1

a₁₄ = a₉ + 5d

33 = 23 + 5d

10 = 5d

2 = d

a_n = a_n-1 + 2

Answer 2

If it is an "arithmetic" sequence, that means it is linear....a given change in the index will produce the same change in value.

We see the index changing by 5 units, and the value changing by 10. That is a 2:1 ratio. Thus, if the index goes from 9 to 1 (a drop of 8), the value will decrease by 16 to 7

The formula: N = 7 + 2(i - 1)

CHECK, i = 9 , N = 7 + 2(8) = 23