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Sequences and Series

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

Finite Series


Evaluate \[1 + 2 + 3 + \cdots + 98 + 99 + 100 = \sum_{i=1}^{100} i. \]

Let \[ S_n = \sum_{k=1}^{n} k^2.\]

Which of the following is not equal to \(S_6\)?

These numbers are called centered pentagonal numbers. A finite series can be used to find the answer.

Let \(\displaystyle S_n = \sum_{k=1}^{n} (-1)^{k^2 + k}\). Then \(S_n = \,\)?

The distance from A to B is 10 kilometers. Starting at A, Zeno repeats the following step:

Iterative Step: Walk halfway to B; this point is the starting point for the next iteration.

How many kilometers will Zeno have walked after performing the iterative step 100 times?


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