Question

Write the recursive rule and the iterative rule for the following sequence: 10, 6, 2, -2, -6

Answer 1

The recursive rule is
$a_n=a_{n-1}+(-4)$. 
The iterative rule is
$a_n=14-4n$

The recursive rule is given by the formula 
$a_n=a_{n-1}+d$, where d is the common difference.  Our common difference is -4, which gives us the recursive rule above.

The iterative rule begins with the formula
$a_n=a_1+d(n-1)$, where a₁ is the first term and d is the common difference.  Our first term is 10 and our common difference is -4:
$a_n=10+(-4)(n-1) \\\text{Using the Distributive Property,} \\ \\a_n=10+(-4*n)+(-4*-1) \\ \\a_n=10-4n+4 \\ \\\text{Combining like terms,} \\ \\a_n=14-4n$