Question

Find an equation for the nth term of the arithmetic sequence.

a10 = 32, a12 = 106

an = -301 + 37(n - 1)
an = -301 + 37(n - 2)
an = -301 + 37(n + 1)
an = -301 - 37(n + 1)

Answer 1

`a_n=-301 +37(n-1)`

Explanation:

arithmetic sequence formula:
`a_n=a +(n-1)d`

where
`a` is the first term and
`d` is the common difference

Given:

`a_(10)=32`

⇒
`a +(10-1)d=32`

⇒
`a+9d=32`

Given:

`a_(12)=106`

⇒
`a +(12-1)d=106`

⇒
`a+11d=106`

Rearrange the first equation to make
`a` the subject:

a = 32 - 9d

Now substitute into the second equation and solve for
`d`

(32 - 9d) + 11d = 106

⇒ 32 + 2d = 106

⇒ 2d = 106 - 32 = 74

⇒ d = 74 ÷ 2 = 37

Substitute found value of
`d` into the first equation and solve for
`a`:

a + (9 x 37) = 32

a + 333 = 32

a = 32 - 333 = -301

Therefore, the equation is:
`a_n=-301 +37(n-1)`