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68 NOTES ON ALGEBRA.
Length = width + depth, =6+4, =10 feet.
This satisfies the conditions of the problem, because
X 4 X 10 = 240 cubic feet,
and the width and depth added together equal the length.
Progressions.
A progression is a series of numbers increasing or decreas-
ing, according to a fixed law.
The successive numbers of which the progression consists
are called terms j the first and the last terms are called the
extre?nes and the others are called the means.
ARITHMETICAL PROGRESSION.
An arithmetical progression is a series of numbers which
increase or decrease by a constant difference. For instance :
2, 4, 6, 8, 10, 12, 14, 16, is an ascending series ;
20, 18, 16, 14, 12, 10, S, is a descending series.
In each of these series the common difference is 2.
The following are the elements considered in an arith-
metical progression
:
a = First term ; / = last term ; d = the common difference ;
11 = the number of terms ; s = the sum of all the terms.
When any three of these quantities are known the other two
may be calculated
:
In the above example of an ascending series :
a = 2; I = 16 ; d= 2 ; n = 8 ; s = 72.
Formulas
:
Examples
:
a = l—{n — l)Xd a — 16 — (8 —1) X 2 = 2
/= a + (u — 1) X d 1=2 + (S — 1) X 2 = 16
d- l ~ a
n — 1
d- 1Q ~ 2 -2
8 — 1
/
—
a i_ 1
n — —j- + 1
d
,-16-2 +1 _ 8
2
r
-(a + l)Xn
2
s
- (2 + 16) X 8 _ 72
2
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