Question

What is the pattern in the sequence 100, 19, 83, 34, 70, 45?

Answer 1

`a_0 = 100`
`a_n = a_(n-1) + (-1)^n*(10-n)^2`

or

`a_(2n) = 100-19n+2n^2`
`a_(2n+1) = 19+17n-2n^2`

Explanation:

(See image below. I couldn't figure out an easy way to show this with standard text)

If you look at differences of alternate terms, you can find the formula for term n as follows:

`100, 83, 70 -> -17, -13 -> 4`

Hence `a_(2n) = 100 -17n +4(n(n-1))/(2!) = 100-19n+2n^2`

`19, 34, 45 -> 15, 11 -> -4`

Hence `a_(2n+1) = 19 +15n -4(n(n-1))/(2!) = 19+17n-2n^2`

See: http://socratic.org/questions/how-do-you-find-the-general-term-for-a-sequence