14. Sequences & Series

The first 4 terms of a sequence are $\left\lbrace\sqrt3,2\sqrt3,3\sqrt3,4\sqrt3,\ldots\right\rbrace$. Continuing this pattern, find the $7^{\th}$ term.

Determine the first 3 terms of the sequence given by the general formula

$a_{n}=\frac{1}{n!+1}$

Write the first 6 terms of the sequence given by the recursive formula $a_{n}=a_{n-2}+a_{n-1}$ ; $a_1=1$ ; $a_2=1$.

Write a recursive formula for the arithmetic sequence.

$\left\lbrace8,2,-4,-10,\ldots\right\rbrace$

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

e.The sequence aₙ = n² / (n² + 1) converge.

a.Does the sequence { k/(k + 1) } converge? Why or why not?

Suppose the sequence { aₙ} is defined by the explicit formula aₙ = 1/n, for n=1, 2, 3, .....Write out the first five terms of the sequence.

Suppose the sequence { aₙ} is defined by the recurrence relation a₍ₙ₊₁₎ = n · aₙ , for n=1, 2, 3 ...., where a₁ = 1. Write out the first five terms of the sequence.