**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, . . . :

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.

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?*

*A. 90
B. 125
C. 200
D. 285
E. 682*

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

The answer is (D).

Example

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

*A. 60
B. 116
C. 210
D. 263
E. 478*

<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).