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

Jul 23, 2015

${a}_{0} = 100$
${a}_{n} = {a}_{n - 1} + {\left(- 1\right)}^{n} \cdot {\left(10 - n\right)}^{2}$

or

${a}_{2 n} = 100 - 19 n + 2 {n}^{2}$
${a}_{2 n + 1} = 19 + 17 n - 2 {n}^{2}$

#### 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:

$100 , 83 , 70 \to - 17 , - 13 \to 4$

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

$19 , 34 , 45 \to 15 , 11 \to - 4$

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