Adding 0 to the first 1 gives 1; adding 1 and 1 gives

2, and then 1 and 0 are 1.

the second power are 1, 2, 1.

Hence the coefficients of

Again, 0+1=1; 1 + 2 = 3; 2 +1=3; and 1+0=1. Hence 1, 3, 3, 1 are the coefficients of the third power.

Again, 0+1=1; 1 + 3 = 4; 3+3=6; 3+1=4; and 1+0=1. Hence 1, 4, 6, 4, 1 are the coefficients of the fourth power.

Again, 0+1=1; 1+4 = 5; 4+6 = 10; 6+4 = 10; 4+1 = 5; and 1+0= 1. Hence 1, 5, 10, 10, 5, 1 are the coefficients of the 5th power, &c.

1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1

Here we observe that the first row of figures taken obliquely downward is the series of numbers 1, 1, 1, &c.

The second row is the series of natural numbers, 1, 2, 3, 4, 5, &c. whose differences are 1.

The third row is the series 1, 3, 6, 10, 15, &c. whose differences are the last series viz. 1, 2, 3, 4, &c.

The fourth row is the series 1, 4, 10, 20, 35, &c. whose differences are the last series viz. 1, 3, 6, 10, &c. Each successive row is a series, whose differences form the preceding row.

We may observe farther that the coefficient of the second term of any power is the term of the series 1, 2, 3, 4, &c. denoted by the exponent of the power. That of the second power is the second term; that of the third power, the third term; that of the nth power, the nth term. But this being the series of natural numbers, the number which denotes the place of the term is equal to the term itself, so that the coefficient of the second term will always be equal to the exponent of the power.

The coefficient of the third term of any power is the term of the series 1, 3, 6, 10, &c. denoted by the exponent of the power diminished by 1. That of the third power is the second term, that of the fourth power the third term, that of the nth power the (n-1)th term, &c.

The coefficient of the fourth term of any power is the term of the series 1, 4, 10, 20, &c. denoted by the exponent of the power diminished by 2. That of the fourth power is the second term, that of the fifth power

is the third term, that of the nth power is the (n-2)th term. And so on as we proceed to the right, the place of the term in the series is diminished by 1.

We may observe another remarkable fact, the reason of which will be manifest on recurring to the formation of these series. We shall take the 7th power for an example, though it is equally true of any other.

The coefficient of the second term, viz. 7, is the sum of 7 terms of the preceding series 1, 1, 1, &c. and was in fact formed by adding them.

The coefficient of the third term, 21, is the sum of the first six terms of the preceding series, 1, 2, 3, &c. and was actually formed by adding them, as may be seen by referring to the formation.

The coefficient of the fourth term, 35, is the sum of the first five terms of the preceding series, 1, 3, 6, 10, &c. and was formed by adding them.

The same law continues through the whole. If now we can discover a simple method of finding the sums of these series without actually forming the series themselves, it will be easy to find the coefficients of any power without forming the preceding powers. This will be our next inquiry.

XLII.

Summation of Series by Differences.

It is not my purpose at present to enter very minutely into the theory of series. I shall examine only a

few of the most simple of them, and those principally with a view of demonstrating the binomial theorem.

A series by differences is several numbers arranged together, the successive terms of which differ from each other by some regular law.

I call a series of the first order that, in which all the terms are alike, as 1, 1, 1, 1, &c. 3, 3, 3, 3, &c. a, a, a, a, &c. In these the difference is zero.

The sum of all the terms of such a series is evidently found by multiplying one of the terms by the number of terms in the series. Every case of multiplication is an example of finding the sum of such a series. The sums of a number n of terms of any series a, a, a, &c. is expressed

A series in which the terms increase or diminish by a constant difference, is called a series of the second order. As 1, 2, 3, 4, 5, &c. 3, 6, 9, 12, &c. or 12, 9, 6, 3. A series of this kind is formed from a series of the first order. The differences between the successive terms form the series from which it is derived.

At present I shall examine only the series of natural numbers 1, 2, 3, 4,

n.

This series is formed as follows:

0+1=1

1 + 1 = 2

The sum

1+1+1=3

1+1+1+1=4

1 +1 +1 +1 + 1 = 5 &c.

of any number n of terms of the series &c. is equal to the nth term of the series 1, 2, 3, 4, &c.

1, 1, 1, 1,

Write down two of these series as follows and add the corresponding terms of the two together.

(n+1),(n+1),(n+1),(n+1). . .(n+1),(n+1),(n+1),(n+1)

The 6th term of the series is 6, and it appears that 5 times 6 will be twice the sum of 5 terms of the series.

The (n + 1)th term of the series 1, 2, 3, 4, &c. is n+1. It appears that n times (n+1) will be twice the sum of n terms of the series.