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The coefficient of the 6th term is the 3d term of the series of the sixth order, which is the sum of 3 terms of the series of the 5th order. The sum of 3 terms of this series is found by multiplying the 4th term by 3 and dividing the product by 5. The 4th term is the coefficient last found, viz. 35
The coefficient of the 7th term is the 2d term of the series of the 7th order, which is the sum of two terms of the series of the 6th order. The 3d term of this series is the coefficient last found, viz. 21.
The coefficient is 7.
The coefficient of the last term is 1, though it may be found by the rule
Examining the formation of the above coefficients, we observe, that each coefficient was found by multiplying the coefficient of the preceding term by the exponent of the leading quantity a in that term, and dividing the product by the number which marks the place of that term. Thus the coefficient of the third term was found by multiplying 7, the coefficient of the second term, by 6, the exponent of a in the second term, and dividing the product by 2, the number which marks the place of the second term. This will be true for all cases, because that exponent must necessarily show the number of terms of which the sum is to be found; the coefficient will always be
the term to be multiplied, because the number of terms always diminishes by 1 for the successive coefficients, and the place of the term always marks the order of the series of which the sum is to be found.
Hence is obtained the following general rule.
Knowing the coefficient of any term in the power, the coefficient o the succeeding term is found by multiplying the coefficient of t known term by the exponent of the leading quantity in that term, and dividing the product by the number which marks the place of that term from the first. y
The coefficient of the first term, being always 1, is always known. Therefore, beginning with this, all the others may be found by the rule.
It may be farther observed, that the coefficients of the last half of the terms, are the same as those of the first half in an inverted order. This is evident by looking at the coefficients, page 207, and observing that the series are the same, whether taken obliquely to the left or to the right.
It is also evident from this, that a + æ is the same as a + a, and that, taken from right to left, a is the leading quantity in the same manner as a is the leading quantity from left to right.
Hence it is sufficient to find coefficients of one half of the terms when the number of terms is even, and of one more than half when the number is odd. The same coefficients may then be Mouen before the corresponding terms counted from the right.
In the above example of the 7th power, the coeffiélents of the first four terms being found, we may begin on the . and put 7 before the second, 21 before the third, 35 before the fourth, and then the power is complete.
Make a = —b, then having found the 3d power of a + a put — b in the place of a and it becomes
In fact it is evident that the powers of a -b will be the same as the powers of a + b, with the exception of the signs. It is also evident that every term which contains an odd power of the term affected with the sign — must have the sign —, and every term which contains an even power of the same quantity must have the sign +.
Make a-b = m, raise m + c to the 3d power, and then sub stitute the value of m.
which is the same as the last, except that the terms which contain the odd powers of b have the sign —.
Hence it is evident that the powers of any compound quantity whatever, may be found by the binomial theorem, if the quantity be first changed to a binomial with two simple terms, one letter being made equal to several, that binomial raised to the power required, and then the proper letters restored in their places.
XLV. The rule for finding the coefficients of the powers of binomials may be derived and expressed more generally as follows:
It is required to find the coefficients of the nth power of a + r.
It has already been observed, Art. XLI., that the coefficient of the second term of the nth power is the nth term of the series of the second order, 1, 2, 3, &c., or, the sum of n terms of the series 1, 1, 1, &c.; that the coefficient of the third term is the sum of (n — 1) terms of the series of the second order; that the coefficient of the fourth term is the sum of (n − 2) terms of the series of the third order, &c. So that the coefficient of each term is the sum of a number of terms of the series of the order less by one, than is expressed by the place of the term ; and the number of terms to be used is less by one for each succeeding series.
By Art. XLII. the sum of n terms of the series 1, 1, 1, is ; The sum of (n − 1) terms of the series of the second order is n (n − 1). 1 × 2. The sum of (n — 2) terms of the series of the third order is n (n − 1) (n − 2) 1 × 2 × 3
It may be observed that n is the exponent of a in the first
term, and that n or its equal # forms the coefficient of the second term. 1