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Sequences and Series

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

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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