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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?

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 – 1) + (2x – 2) + ... + 2 + 1. What is the value of <3> × <2> ?

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

 

<3> = 2(3) + (2 × 3 – 1) + (2 × 3 – 2) + (2 × 3 – 3) + (2 × 3 – 4) + (2 × 3 – 5)
= 6 + 5 + 4 + 3 + 2 + 1 = 21

 

<2> = 2(2) + (2 × 2 – 1) + (2 × 2 – 2) + (2 × 2 – 3)
= 4 + 3 + 2 + 1 = 10

 

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





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