Question

Enter the recursive rule for the geometric sequence. screenshot included!

Answer 1

Explicit rule:
a(n)=(2/5)(5^(n-1))

For a recursive rule, we need to express a(n) in terms of a(n-1), which we can obtain from the explicit rule
a(n)=(2/5)(5^(n-1))
substitute n-1 for n above
a(n-1)=(2/5)(5^((n-1)-1))
=(2/5)(5^n-2)

Divide:
a(n)/a(n-1)=(2/5)(5^(n-1)) / ((2/5)(5^(n-2)))
=1/5^(-1)
=5

Therefore, multiplying both sides by a(n-1)
a(n)=5 a(n-1)
a(1)=(2/5)(5^(1-1))=2/5

So the recursive rule is
a(1)=2/5, a(n)=5 a(n-1)