find a formula an, for the n-th term of the following sequence. assume the series begins n=1. 1/1, -1/8 1/27... ( use symbolic notion and fractions where needed)

Answer 1

In Algebra, we usually check for an Arithmetic Sequence (common difference), then for a Geometric Sequence (common ratio). This sequence is neither.

With only three terms, there may be many, many formulas that produce them.

If we notice (1) alternating + and -, and (2) denominators getting large rapidly, we may write the explicit formula:

a_n = (-1)^n-1 / n³ for n>0

To check:

a₁ = (-1)^1-1 / 1³ = (-1)⁰ / 1³ = 1 / 1

a₂ = (-1)^2-1 / 2³ = (-1)¹ / 2³ = - 1 / 8

a₃ = (-1)^3-1 / 3³ = (-1)² / 3³ = 1 / 27