(18.) Corollary. Hence, in a series of squares, or second powers, the second differences are equal; in a series of cubes, or third powers, the third differences are equal, and so on.

Examples

1. Find the second order of the differences of the series of squares 1o, 2o, 3o, 4o, 5o, &c.

Here n = 23212.

So that the second order of differences are a series of two's, as will be found by actually taking the differences of the numbers themselves, thus;

2. Find the third order of differences of the series of cubes 13, 23, 3',4', &c. that is of 1, 8, 27, 64, &c.

=

Here 3321=6, as will be found by actually taking the differences of the numbers themselves, thus

Application of the direct method of differences, to the finding of any term of a series, or any term of any order of differences of a series

Definitions.

1 A series is a rank of numbers or quantities succeeding each other according to a certain law.

2 General term, of a series is such a function of a quantity that if the number which indicates the number of the term of the series be substituted for that quantity, the function will coincide with the term itself.

Thus if the series be 3, 7, 11, 15, &c. and let 4x-1 be a function of, then to ascertain whether this function be the general term; let be put equal to any particular value as 4, then the value of the function 4r-1 will become 16-1=15, which coincides with the 4th term of the series; and if this coincidence take place in every similar supposition, then will 4x-1 be the general term of the series; or by making a respectively equal to 1, 2, 3, &c. we shall be enabled to continue them at pleasure. This function 4 x −1 is the law which regulates this particular series, and so on in other instances.

3. The first order of differences of a series, is another series of numbers found by subtracting the first term of the given series from the second, the second from the third, the third from the 4th, &c. The second order of differences is a third series found by subtracting the first term of the first order of differences from the second, the second from the third, the third from the 4th, and so on.

Corrollary. Hence, because every two consecutive terms of a series in any order of differences, may be respectively represented by the ith term and the (x+1)th term; that is, by the same functions of r and x+1, it is evident that the difference of the two consecutive terms themselves of the series must be the same as the difference between their respective general terms, when the quantity r is made to coincide with the number of the term of the series.

Problem 9.

(19.) Given the first term of a series of arithmetical progressionals, and the common difference of the terms, to find a general expression for any term.

Let r be the number of the term, then, the general term will be found by the following

Rule.

(20.) Multiply the common difference into (x-1) and add the first term to the product, then the sum will be the general term.

Example 1. Find the rth term of the series 1, 4, 7, 10, &c. Here the common difference is 3, and the first term is 1; therefore the general term is 3(x-1)+1=3x−3+1=3x-2.

Example. Find the rth term of the general series*

Here

a, (a+c), (a+2c), (a + 3c), 2, &c.

is the common difference, and a is the first term, whence by rule c(x-1)+a is the general term.

Problem 10.

(21.) To find the general term of a series of which all the terms are composed of the same number of factors, and of which the difference

A symbol representing any series of arithmetical progressionals may be thus represented a rlc as in factorials.

between every two corresponding factors of every two consecutive terms, is equal to the same given number.

Rule.

(22.) Find a general expression for each of the factors, as in the last Problem, then indicate the product of these factors, and the form of the product will express the general term.

Example 1. Find the general term of the series

3.3+4.5+5.7+, &c.

Here the progressions formed of each series of factors are 3, 4, 5, &c. and 3, 5, 7, &c; now, by the first of these series, we have 1(x-1)+3x+2; and, by the second, we have 2(x-1)+3=2x+1; wherefore, the general term will be represented by (+2) (2x+1). Example 2. Find the general term of the series

1.45 +3.7.9+5.10.13+7.13.17+, &c.

Here the progressions formed of each series of factors are respectively, 1, 3, 5, 7, &c. and 4, 7, 10, 13, &c. and 5, 9, 13, 17, &c.; now, by the first of these, we have 2(x-1)+1=2x−1; by the second, 3(x-1)+4=3x+1; and, by the third, 4(x−1)+5=4x+1; whence the general term will be represented by (2x-1) (3x+1)(4x+1).

3. Find the general term of the series r3+r7+7+, &c. Here the series of exponents are 5, 7, 9, 11, &c.; therefore, the general exponent is 2(x-1)+5=2x+3; whence the general term is 72*+3.

Problem 11.

A series, consisting of integral factors being given, to find any term of any order n of differences.

Rule.

(23.) Find the general term of the series by Problem 10; then, if it be a factorial, find the nth order of differences by Problem 4; but if it is not, resolve it into factorials by Problem 1; then find the nth order of differences of each of the terms that have their exponents equal to, and greater than, the number expressing the number of differences; then make x equal to the number of the term required.

Examples.

1. Find the third term of the first order of differences of the series 42, 52, 67, 7°, &c.

Here the general term is (x+3), which being reduced to factorials, of which the common difference is unity, the difference of x, it wil become x2+5x+9.

By rule (x+3)=x2+5x=2(x+1)''' +111ī.5(x+1)¤1

=2(x+1)+5=2x+7.

2. Find the 20th term of the 4th order of differences of the series 12.23, 29.39, 32.49, 42.59, &c.

Then finding the 4th order of difference of the first two terms, as the exponents of the other terms are less than 4, we shall have,

=5.4.3.2(x+4)-7.4.3.2.1.

Then making r=20, this expression becomes,

5.4.3.2.24-7.4.3.2.1=2712.

Problem 12.

A series, consisting of the reciprocals of integer factors being given to find any term of any order n of differences.

Rule.

24. Find the general term of the series by Problem 10; and if it be a factorial reciprocal, find the nth order of differences by Problem: 6; but if it is not, resolve it into factorial reciprocals by Problem 2: then find the nth order of differences of each of the terms by Problem 6; and make x equal to the number of the term.

Example.

Find the first term of the fourth order of differences of the series 1 1 1

1

1

&c.

3.4' 4.5' 5.6' 6.7' 7.8'

Here the general term of this series is

=(x+2)−211,

(x+2)(x+3)=(x+2)21:

Therefore, by rule A(x+2)−11x − (−2) 41ī‍× (x+2—2—41—.

then, by making r equal to 1, we shall have,

,, &c. being given to find any

Rule

25. Find the general term of the series by Problem 10 and

multiply (——) )* into the general term; then make a equal to the

number of the term, and the result will be the number of the term required.

Examples.

1. Find the first term of the fifth order of differences of the 1 1 1 1 1

series

&c. 1, 2, 22, 29, 24'

52 A

By rule ▲ 2 = ( -—-—1)' × 2 will express any term of the 5th

2x

order of differences; therefore, making r equal to 1, and we snall

2. Find the second term of the third order of differences of the series

3 1

Whence by rule = ( − 1 ) ' × 1 − ( − 1 ) 3 × 1

A

3*

X

3*

Then.making x = 2 then 321-33=-283

3*+3 35

243'

Inverse Method of Differences.

Definition.

1. Inverse method of differences, is that of finding the original function from the difference being given.

2. The function thus found is called the integral, the operation is called integration, and the given difference is said to be integrated.

Notation.

The sign of the operation of integration is formed by placing the Greek letter 3 before the quantity which is to be integrated.

As Functions admit of various orders of differences; it is evident that if any function is given as a difference, various orders of inverse operations, may also be found at pleasure. Therefore if an integral is found from the given difference a first time, the sign of integration is formed by placing ▲ before the quantity to be integrated; and if the integral thus found be considered as the difference of another function, the sign of integration will be formed by placing A before the

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function last found, or A before the first given difference, and so on.