Calculus

# Finding the nth Term

Guess the next term of the following sequence:

$2, \quad 3, \quad 5, \quad 8, \quad 12, \quad 17, \quad \square$

What is the $$6 ^{th}$$ term of the sequence $$\{ n^2 + 2 n + 8 \}_{n=1}^\infty?$$

The first $$n$$ terms of a sequence sum to $$n(n+2)(2n+1)$$. What is the value of the $$6 ^{th}$$ term?

The sequence $$\{a_n\}$$ satisfies $$a_1=1$$ and $a_{n+1}=11 ^n a_n \text{ for } n \geq 1.$

For what value of $$k$$ do we have $$a_k=11 ^{276}$$?

If $$\{a_n\}$$ is an arithmetic progression with $$a_{11}=133$$ and $$a_{23}=313$$, what is $$a_{50}$$?

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