Question

How do you find the nth term of the following sequence:? `9, 4, -5, −18, −35`

Answer 1

`a_n = -2n^2+n+10`

Explanation:

Let's look at the differences between successive pairs of terms.

Write down the given sequence:

`color(blue)(9), 4, -5, -18, -35`

Write down the sequence of differences between successive terms:

`color(blue)(-5), -9, -13, -17`

Write down the sequence of differences of those differences:

`color(blue)(-4), -4, -4`

Having reached a constant sequence, we can use the first term of each of the above sequences as coefficients in a formula for the `n`th term `a_n`:

`a_n = color(blue)(9)/(0!)+color(blue)(-5)/(1!)(n-1)+color(blue)(-4)/(2!)(n-1)(n-2)`

`color(white)(a_n) = 9-5n+5-2n^2+6n-4`

`color(white)(a_n) = -2n^2+n+10`