Page 34 - Standard Handbook Of Petroleum & Natural Gas Engineering

Algebra 23

1 (l+x)-Y* =1--x+-x2--xxJ+-x4 35 -
1
14
2
G= 3 9 81 243 . . . (Ixl< 1)
3 3 1 3
16
8
~~=(1+X)"=l--x+-x~--xxJ+-x4-. . . (Ixl< 1)
128
2

with corresponding formulas for (1 - x)'", etc., obtained by reversing the signs
of the odd powers of x. Also, provided lbl < (a(:

(a+b)" =an(l+$)" =a"+nla"-'b+n2a"-2b2+n,a"-'bb"+ . ..

where nl, n2, etc., have th values given above.

Progressions
In an arithmetic progression, (a, a + d, a + 2d, a + 3d, . . .), each term is obtained
from the preceding term by adding a constant difference, d. If n is the number
of terms, the last term is p = a + (n - l)d, the "average" term is 1/2(a + p) and
the sum of the terms is n times the average term or s = n/2(a + p). The arithmetic
mean between a and b is (a + b)/2.
In a geometric progression, (a, ar, ar2, ar', . . .), each term is obtained from the
preceding term by multiplying by a constant ratio, r. The nth term is a?', and
the sum of the first n terms is s = a(r" - l)/(r - 1) = a(l - rn)/(l - r). If r is a
fraction, r" will approach zero as n increases and the sum of n terms will approach
a/( 1 - r) as a limit. The geometric mean, also called the "mean proportional," between
a and b is Jab. The harmonic mean between a and b is 2ab/(a + b).

Summation of Series by Difference Formulas

a,, a2, . . ., an is a series of n numbers, and D' (first difference), D" (second
difference), . . . are found by subtraction in each column as follows:
a D' D" D"' D""
-26
2 28 -16
12
14 3 -9 0
17 -2 0
1
18 5 0
6
24 18 12
42