Defined Geometric Sequence

Question

Consider the sequence defined as a_n=n*a_n-1, a₁=1

a) Write out the first 5 terms (a₁ through a₅) for this sequence.

b) Write an expression that gives the general nth term, a_n for the sequence in a non-recursive form.

Answer 1

a)

a₁=1

a₂=2·1=2

a₃=3·2=6

a₄=4·6=24

a₅=5·24 = 120

b)

a₂=2·a₁ = 2·1

a₃=3a₂=3·2·1

a₄=4a₃=4·3·2·1 See the pattern?

a_n=n(n-1)(n-2)(n-3)...2·1 = n!