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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 of the succeeding term is found by multiplying the coefficient of the 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.
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 275, 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 + x is the same as x + a, and that, taken from right to left, x 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 written before the corresponding terms counted from the right.
In the above example of the 6th power, the coefficients of the first four terms being found, we may begin on the right and put 6 before the second, and 15 before the third, and then the power is complete.
1. What is the 7th power of a + wc? Ans. a? +7 a x +21 a 2 + 35 a 2 + 35 ax + 21 a 2
+7 ax + ".
2. What is the 10th power of a + x? Ans. a"' + 10 ao x + 45 ao xo + 120 a' x + 210 ao x* t... 252 a' x + 210 a* x + 120 aox? + 45 ao 21 + 10 a x' + x". 3. What is the 9th power of a + b? 4. What is the 13th power of m+n? 5. What is the 2d power of 2 ac+d? Make 2 ac=b. The 2d power of b + d is b? + 2 bd + d'.
Putting 2 ac the value of b into this, instead of b, observing that b = 4 a* c, and it becomes
4 a' c +4 acd+d. 6. What is the 3d power of 3 c +2bd? Make a = 3c and x = 26 d. The 3d power of a + is a + 3 a* r -+ 3 a x + x. Put into this the values of a and 2 and it becomes
27 c + 54 c* bd + 36 c* bo d' + 363 d®, which is the 3d power of 3 c* + 2bd.
7. What is the 3d power of a -b?
Make a =-b, then having found the 3d power of a ti put — b in the place of x and it becomes
a.- 3a'b + 3 ab-h, which is the 3d
of a -b. 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 +.
8. What is the 7th power of m - n?
11. What is the 3d power of a + b +c?
mø = 63 + 36 c +3 bc + co. Substituting these values of m, the third power of a +b+c will be a* +3a2b+ 3a*c+3 abi +6 abc +3 acé +63 +36%c +36c*+c'.
12. What is the 3d power of a -6+c? Make a
-b = m, raise m + c to the 3d power, and then substitute the value of m.
Ans. a* — 3 ao b + 3 ac + 3 ab'— 6 abc— 3 a c — 23.... +35c-3bc+c; 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.
13. What is the 2d power of a +b+c-d? Ans. a* + 2 ab +62 +2 ac+2bc-2ad-2bd +c .....
-2cd+d. 14. What is the 3d power of 2 a - b + c ? 15. What is the 7th power of 3 ao
of 3 a 2 a' d? 16. What is the 4th power of 7 62 +2c- do? 17. What is the 13th power of a' — 2 6*? 18 What is the 5th power of a --- 2 d? 19. What is the 3d power of a
of a – 2d+cd? 20. What is the 3d power of a - b— 2 c - do? 21. What is the 5th power of 7 a 18 — 10 ac??
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 + x.
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 i The sum of (n − 1) terms of the series of the second order is
n (n -1).
1 X 2 The sum of (n − 2) terms of the series of the third order is
n (n — 1) (n — 2).
1 X2 X 3 Hence (a + x)"
n (n - 1) = a* + n=+*+
1 X 2
2013 + &c. 1 X 2 X 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.
The coefficient of the third term is a multiplied by "=1,
2 multiplied by (n − 1) and divided by 2. But (n 1) is the exponent of a in the 2d term, and 2 marks the place of the second term from the left. Therefore the coefficient of the third term is found by multiplying the coefficient of the second term by the exponent of a in that term, and dividing the product by the number which marks the place of that term from the left.
By examining the other terms, the following general rule will be found true.
Múltiply the coefficient of any term by the exponent of the leading quantity in that term, and divide the product by the number that marks the place of that term from the left
, and you will obtain the coefficient of the next succeeding term. Then diminish the exponent of the leading quantity by 1 and increase that of the other by 1 and the term is complete.
By this rule only the requisite number of terms can be obtained. For or”, which is properly the last term of (a + x)", is the same as ao x". If we attempt by the rule to obtain another term from this, it becomes 0 x a' x nt) which is equal to
It has been remarked above, that the coefficients of the last half of the terms of any power, are the same as those of the first reversed. This may be seen from the general expression :
This furnishes the following fractions, viz.
į, , , , , , ,