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What's the sum of the first 100 positive integers? How about the first 1000?

Guess the next term of the following sequence:

\[ 2, \quad 3, \quad 5, \quad 8, \quad 12, \quad 17, \quad \square \]

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

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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 \(a_k=11 ^{276}\)?

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