Determine the nth term formula for each of the following sequences and find the limit (if it exists).

Question

Determine the nth term formula for each of the following sequences and find the limit (if it exists).

1. 4,-9, 16,-25, …
2. 1,-1, 1,-1, …

Answer 1

1) a_n = (-1)ⁿ⁺¹(n+1)²

|a_n+1| > |a_n| so the limit DNE. The sequence at n = ∞ can either be +∞ or -∞

2) a_n = (-1)ⁿ⁺¹

a_n oscillates between 1 and -1 and never settles to a constant, so the limit DNE because the sequence must converge to a constant.