# Series

A series is simply the sum of the terms of a sequence. The following is a series of even numbers formed from the sequence 2, 4, 6, 8, . . . :

2 + 4 + 6 + 8 + . . .

A term of a series is identified by its position in the series. In the above series, 2 is the first term, 4 is the second term, etc. The ellipsis symbol (. . .) indicates that the series continues forever.

Example

The sum of the squares of the first n positive integers  is . What is the sum of the squares of the first 9 positive integers?

1. 90
2. 125
3. 200
4. 285
5. 682
Solution

We are given a formula for the sum of the squares of the first n positive integers.
Plugging n = 9 into this formula yields

Example

For all integers x > 1, <x> = 2x + (2x â€“ 1) + (2x â€“ 2) + ... + 2 + 1. What is the value of <3> Ã— <2> ?

1. 60
2. 116
3. 210
4. 263
5. 478
Solution

<3> = 2(3) + (2 Ã— 3 â€“ 1) + (2 Ã— 3 â€“ 2) + (2 Ã— 3 â€“ 3) + (2 Ã— 3 â€“ 4) + (2 Ã— 3 â€“ 5)
= 6 + 5 + 4 + 3 + 2 + 1 = 21

<2> = 2(2) + (2 Ã— 2 â€“ 1) + (2 Ã— 2 â€“ 2) + (2 Ã— 2 â€“ 3)
= 4 + 3 + 2 + 1 = 10

Hence, <3> Ã— <2> = 21 Ã— 10 = 210, and the answer is (C).