Page 35 - Standard Handbook Of Petroleum & Natural Gas Engineering

24 Mathematics

If the kth differences are equal, so that subsequent differences would be zero,
the series is an arithmetical series of the kth order. The nth term of the series
is an, and the sum of the first n terms is Sn, where
an = a, + (n - 1)D' + (n - l)(n - 2)D'y2! + (n - l)(n - 2)(n - 3)D'"/3! + . . .
Sn = na, + n(n - 1)D'/2! + n(n - l)(n - 2)DfY3! + . . .

In this third-order series just given, the formulas will stop with the term in D"'.
Sums of the First n Natural Numbers
To the first power:
1 + 2 + 3 +. . . + (n - 1) + n = n(n + 1)/2
To the second power (squared):
l2 + 22 + . . . + (n - 1)* + n2 = n(n + 1)(2n + 1)/6
To the third power (cubed):
1' + 2' + . . . + (n - 1)' + nJ = [n(n + 1)/212

Solution of Equations in One Unknown
Legitimate operations on equations include addition of any quantity to both
sides, multiplication by any quantity of both sides (unless this would result in
division by zero), raising both sides to any positive power (if k is used for even
roots) and taking the logarithm or the trigonometric functions of both sides.
Any algebraic equation may be written as a polynomial of nth degree in x of
the form
aOxn + a,x"" + a2xn-2 + . . . + an_,x + an = 0
with, in general, n roots, some of which may be imaginary and some equal. If
the polynomial can be factored in the form

(x - p)(x - q)(x - r) . . . = 0
then p, q, r, . . . are the roots of the equation. If 1x1 is very large, the terms
containing the lower powers of x are least important, while if 1x1 is very small,
the higher-order terms are least significant.
First degree equations (linear equations) have the form
x+a=b
with the solution x = b - a and the root b - a.
Second-degree equations (quadratic equations) have the form
ax2 + bx + c = 0
with the solution

-b+Jb2-4ac
x=
2a