nth term algebra

Question

Use a system of two equations in two variables, a, and d, to write a formula for the general term (the nth term) of the arithmetic sequence whose seventh term, a7, is 12 and whose ninth term, a9, is 18.

an= ??

Answer 1

a₇ = a₁ + (n - 7)d

12 = a₁ + 6d, equation 1

18 = a₁ + 8d, equation 2

6 = 2d, equation 2 minus equation 1

3 = d

Substitute d into either equatio 1 or 2 to determine value of a₁

Answer 2

a7=12, a9=18

d = (a9-17)/2 = 18-12)/2 = 3 = common denominator

a7=a1+3(7-1) = a1+18

a1 = a7-18=12-18=-6

an = a1+d(n-1)

an= -6+3(n-1)

an = -6+3n-3

an = 3n-9

-6,-3,0,3,6,9,12,15,18, 21, 24, 27, 30, ..., 3n-9, ...