of the kind, whatever be the original series, and whatever be the first terms of those formed from it. XLIV. Binomial Theorem. Before reading this article, it is recommended to the learner to review article XLI. Let it now be required to find the 7th power of a + x." The letters without the coefficients stand thus ; a’, a® x, ao x®, a* x*, ax*, a' xo, a x®, cc?. The coefficient of the first term we observed Art. XLI, is always 1. That of the second term is 7, the exponent of the power, or the 7th term of the series 1, 2, 3, &c. The coefficient of the third term is the sixth term of the series of the third order 1, 3, 6, 10, &c. which is the sum of six terms of the series 1, 2, 3, &c. This sum is found by multiplying the 7th term of the series by 6 and dividing the product by 2. But the 7th term is 7, the coefficient last found. 6 x 7 2 The coefficient is 21. The coefficient of the fourth term is the 5th term of the series 1, 4, 10, &c., or it is the sum of five terms of the preceding series. The sum of five terms of the series 1, 3, 6, &c., is found by multiplying the 6th term by 5 and dividing the product by 3. The 6th term is the coefficient last found, viz. 21. = 21. 5 X 21 = 35. 3 The coefficient is 35. The coefficient of the fifth term is the fourth term of the series of the fifth order 1, 5, 15, &c., or it is the sum of 4 terms of the preceding series. The sum of 4 terms of the series 1, 4, 10, &c. is found by multiplying the fifth term of the series by 4 and dividing the product by 4. The fifth term is the coefficient last found, viz. 35. 4 x 35 535. 4 Thé coefficient is 35. 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 3 x 35 = 21. . 5 The coefficient is 21. 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. 2 x 21 = 7. 6 The coefficient is 7. The coefficient of the last term is 1, though it may be found by the rule Hence the 7th power of a + x is a' + 7 a® x +21 a® x2 + 35 a* x* + 35 ao 2* +21 a 2 + 7 a 28 +27 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 of the succeeding term is found by inultiplying 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. Examples. 1. What is the 7th power of a + x? Ans. a' + 7 a® +21 at +35 a + 35 a*** + 21 a * +7ax + x? 2. What is the 10th power of a + x? Ans. a' + 10 ao x + 45 a 20€ + 120 a' x + 210 ao x* t... 252 ao 2% + 210 a* x + 120 ao x? + 45 a® 2.8 + 10 a x + xo. 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 b2 +26d+d". Putting 2 ac the value of b into this, instead of b, observing that 1 = 4 ac, and it becomes 4 ac +4acd+d. 6. What is the 3d power of 3 c + 2bd? Make a = 3c and x = 26 d. The 3d power of a + x is a' + 3 a* x + 3 a 2 + x. Put into this the values of a and x and it becomes 27 C + 54cbd + 36 c*b* d +86d", which is the 3d power of 3 c + 2bd. 7. What is the 3d power of a ---b? Make x=-b, then having found the 3d power of a to put - b in the place of x and it becomes ai - 3 ab + 3 ab— 69, which is the 3d power 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? m = 68 + 36 c +3 b c + cs. Substituting these values of m, the third power of a +b+c will be a'+3a'b+ 3a'c+ 3 ab*+6 abc +3 ac +6+36 c +360* +c. 12. What is the 3d power of a -6+c? Make a — b = m, raise m tc to the 3d power, and then substitute the value of m. Ans. as - 3 ab + 3 ac +3 ab- 6abc-3ac-63.... +36-3 b c +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— 2 ad— 262 +c..... --2cd+d". 14. What is the 3d power of 2 a - b + c ? 15. What is the 7th power of 3 a — 2 a d? 16. What is the 4th power of 7 b3 + 20-d? 17. What is the 13th power of a — 2 b? 18. What is the 5th power of a' — -2 d? 19. What is the 3d power of a — 2 2 + c d ? 20. What is the 3d power of a - b - 26-d’? 21. What is the 5th power of 7 ao bo — 10 a* c* ? |