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70 NOTES ON ALGEBRA.
In this example of a descending series:
a = 48
;
/ = 3; r = 2 ; n = 5 ; j = 93.
Formulas
:
Examples :
a = /X r™- 1
a = 3;x2 w = 3X«
1- a /_ 48 _ 48 _ 48
n-l
/
2 5-1
24 16
5-1 4
/tfp-. 4S — /^: 3
72 = * 2 _|_ J =
* "
/^T7 + * 1.681241 — 0.477121 i
j _g
0.30103
n
aXr—l 48 X 2 —3
The Arithmetical Mean.
The arithmetical mean of two or more quantities is obtained
by adding the quantities and dividing the sum by their number.
For instance, the arithmetical mean of 14 and 16 is
14 + 16
_ _ ^
Thus : The arithmetical mean is simply the average.
The Geometrical Mean.
The geometrical mean of two quantities is the square root
of their product. For instance, the geometrical mean of 14 and
16 is Vl4 X 16 = 14.9666.
The geometrical mean of two numbers is also called their
mean proportional.
When the difference between two numbers is small as com-
pared to either of them, their arithmetical mean is approxi-
mately equal to their geometrical mean.
This fact may be used to advantage for calculating approxi-
mately a root of any number.
For instance, find the square root of 148.
Knowing that the square of 12 is 144, twelve is used as a
divisor, thus
:
^= 12.333, and
12- 883
/ 12
= 12.166,
12 2
which is correct within 0.005.
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