Question

How do you write a rule for the nth term of the arithmetic sequence given `a_10=8`, `a_16=32`?

Answer 1

`{(a_0=-32),(a_(i+1)=a_i+4):}`

Explanation:

For an arithmetic sequence `< a_0, a_1, a_2, a_3, ... >`
terms are related by the formula:
`color(white)("XXX")a_m=a_n + (m-n) * k` for some constant `k`

In this example:
`color(white)("XXX")a_16=a_10+(16-10) * k`
or (using the given values)
`color(white)("XXX")32=8+6 * k`

`color(white)("XX")rarrk=4`

And the initial value, `a_0` is
`color(white)("XXX")a_0 = a_10+(0-10) * k`
or
`color(white)("XXX")a_0 = 8 +(-10) * (4) = -32`

If your standard is to use `a_1` as an initial value:
`color(white)("XXX")a_1=a_10+(1-10) * k`
`color(white)("XXXXX") = 8 +(-9) * (4) = -28`