THEOREMS. 366. If a, b, c, are in H. P., then a c = a − b b — c. 367. The harmonic mean between two numbers is twice their product divided by their sum. Let a, b, be the two numbers, and H their harmonic mean. 368. The geometric mean between two numbers is the geometric mean between the arithmetic and harmonic means of the same numbers. Let A, G, H, represent, respectively, the arithmetic, geometric, and harmonic means between a and b; then 369. EXAMPLES. 1. The second term of an H. P. is 2, and the fourth term is 6. Find the series. The 2d and 4th terms of the corresponding A. P. are 1 and 1, respectively. Therefore ad = = , and a 3 d a = , and d 1. Therefore the A. P. is, 1, and the H. P. is, 2, 3, 6,... ; whence b + c'a + c' a + b' 1 are in A. P. are in A. P. с a + c 1+ 1 + a + b are in A. P. b с a + c a+b are in A. P. b a + c a+b + c Find the arithmetic, harmonic, and geometric means be 6. Insert three harmonic means between 2 and 12. 7. Continue to five terms the H. P. 13, 1, . Ans. 3, 4, 6. 8. The first term of an H. P. is unity, the third term is } Find the tenth term. Ans. 9. If b + c, a + c, a + b, are in H. P., prove that a2, b2, c2, are in A. P. 10. The difference of the arithmetic and harmonic means between two numbers is 1. Find the numbers, one being four times the other. Ans. 2, 8. CHAPTER XXIV. THE BINOMIAL THEOREM. 370. THE laws for the expansion of a binomial, as illustrated by actual multiplication in Art. 208, can be proved to be true for any index. The following is the proof when the index is any positive integer. Following the laws of Art. 208, we have Now, multiplying this equation by a + b, for the first member we have (a+b)+1; and for the second, and this is exactly the form that is obtained in applying the laws of Art. 208 directly to expanding a + b to the nth power. Therefore, · if the laws are true for the power whose index is m, they are true for the power whose index is (m + 1). These laws have been proved to be true for the 5th power (§ 208); therefore they are true for the 6th; and therefore for the 7th; and so on. NOTE. This method of proof is called mathematical induction. 371. Ifb is substituted for b, in the binomial a+b, the signs of the terms containing the odd powers of b will be (§ 208); that is, the second, fourth, sixth, etc. terms. 372. To find any term in the expansion of a binomial. The indices in any term in the expansion of (a + b)" can be written at once from the last formula in Art. 370. Thus, the indices in the rth form are, for a, n-(-1)= n+1−r, and for b, r—1; and if the sign of b is +, the sign of the rth term is, but if b is and r 1 is odd, that is, if r is even, the sign is, otherwise +. 6th term is 18 — (6 − 1) = 13; and the index of b is 5. The sign NOTE 1. In writing the formula for the coefficient, write for the denom inator 1234... etc., with the last number =r- 1; then, for the numerator, over each number in the denominator write a series beginning with a number =n, and decreasing by 1. Find the 2. 8th term in (x − y)16. 5. 96th term of (m — n)100. NOTE 2. Notice that the 96th term of the 100th power is the 6th from the last term. NOTE 3. Whether the index is positive or negative, integral or fractional, the Binomial Theorem can be applied equally well. 12. Expand (a + b)−1. (a + b) -1 = a1 + (−1) a−2 b + (1) a−3 b2 + (−1) a−4 b3 +, etc. a-2b+a-3 b2 — a − 4 b3 +, etc. = a (ab)-1= a -1 — (−1) a−2b + (1) a−3 b2 − (− 1) a−b8+, etc. — +a-2b+a-3 b2 + a−4b8+, etc. 1 b b2 b3 =- + + + +, etc. α a2 a3 a4 NOTE 4. The same results will be obtained in these examples if we write (a+b)-1 and (a - b)−1 in their other forms, and and perform the division. a + b a |