Sequences and Series

Finite Series

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

5000 = 50 \times 100

4950 = \frac{99 \times 100}{2}

5050 = \frac {100 \times 101}{2}

2500 = 50\times 50

by Brilliant Staff

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

Which of the following is not equal to $S_6$ ?

\frac{6 \times 7 \times 14}{6}

S_7 - 7^2

S_5 + 6^2

by Brilliant Staff

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

1 + 5(20)

1 + 5\left(\frac{20(21)}{2}\right)

by Brilliant Staff

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

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n

0 or 1, depending on the value of

n

by Brilliant Staff

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?

\sum_{n=1}^{100} 10\left(\frac{1}{2}\right)^n

\sum_{n=1}^{100} (5)^n

by Brilliant Staff