SERIES—SHAPES AND SQUARES 25

numbers from 1 upwards represents a square number, sug-
gests that there may be some similar property in a series
of even numbers. In fact there are several relationships
apparent in such a series.

2 == =22—2
244 -, =3%—3
24446 =12 =4%—4
2+4+64-8 =20 =pj2—p

2+4+6+8+410 =30 =62—6
It will be seen that the sum of any series of even numbers
commencing at 2 and omitting none can be calculated by
adding 1 to the number of terms=(n+1), squaring the re-
sulting number = (n+1)? and taking from this total the same
number (n+1) giving a final result of (n41)%2—(n-+1). Each
total is thus related to the square of the number which ex-
ceeds the number of terms by one.
This leads to another relationship. The expression
(n+1)2—(n+1) is equivalent to (n+1)(n+1—1)
=(n+1)n=n2+n.
Thus the sum of an even number series also equals 7247
as follows:
2 = 2 =1%241
2-+4 = 0 =2%4}2
2+4+6 =12 =3%|3
This relationship can be proved as follows. The sum of an
Arithmetical Progression is 4(a+-L)n. In the particular series

under consideration, « is always 2 and L is always equivalent
to 2n.

o S=L(2+2n)n
_2n+2n?

 

=n-+n?

2
From this same formula, a further relationship arises, for
n-+n*=(n)(n+1) and so the sum of any similar series is equal
to the product of the number of terms (r) and the number
(n+1).
Thus 244+4+6=12=3 X4

 

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