7. The perimeter, a+b+c+d, of a rectangle is 36, and the area of the rectangle is 80. Find the sides. 8. A farmer bought 15 cows and 20 sheep for $720. He bought 3 more cows for $320 b than he did sheep for $30. Find the price of each. a 9. The sum of the numerator and denominator of a fraction is 7. If the numerator be diminished by 1, and the denominator be increased by 1, the product of the resulting fraction and the original fraction is 10. Find the fraction. 10. If 7 be added to the numerator of a fraction the value of the fraction becomes 7. If the square of the denominator be subtracted from the square of the numerator the result is 7. Find the fraction. B 11. The area of a triangle ABC is onehalf the product of the base, AC, and the altitude, DB. The area is 48 square feet. BC is 10 feet and its square is equal to the sum of the squares of BD and DC. AD=DC. Find AC and BD. Can more than one such triangle be drawn? A C D 12. A triangle ABC has the angles B and C equal. The angle A is 60° more than the square of the number of degrees in the angle B. The sum of the three angles is 180°. Find the angles. 13. A travels from C to D. Two hours after he leaves C, B starts out to overtake him, traveling 3 miles per hour faster than A. Had A traveled 1 mile per hour slower, B would have overtaken him 12 miles nearer to C. Find A's rate. 14. In a triangle with a right angle at C, the altitude drawn from C to the hypotenuse is a mean proportional between the segments, a and b, of the hypotenuse. We A b B know also that BC2=h2+b2. If AC=12, CB=9, and AB= 15, find a, b and h. 15. The sum of two numbers is to their difference as 7 is to 2. The ratio of their product is to the product of their sum and difference as 45 is to 56; find the numbers. (Is the statement or the solution the more difficult?) 16. In a right cone, we know from geometry that and = S=TRH, V=TR2A, where S lateral surface, R= radius of base, V = volume, H= slant height, A=altitude. If S 60 and H=10, find V. (Remem = ber that because of the right angle at D, H2 = A2+R2.) XVI. THE BINOMIAL THEOREM POSITIVE INTEGRAL EXPONENT 249. A Series is a succession of terms. A Finite Series is one having a limited number of terms. An Infinite Series is one having an unlimited number of terms. 250. In §§ 91 and 183 we gave rules for finding the square or cube of any binomial. The Binomial Theorem is a formula by means of which any power of a binomial may be expanded into a series. 251. Proof of the Binomial Theorem for a Positive Integral Exponent. The following are obtained by actual multiplication : (a+x)1=a1+4a3x+6 a2x2+4 ax3+x1; etc. In these results, we observe the following laws: 1. The number of terms is greater by 1 than the exponent of the binomial. 2. The exponent of a in the first term is the same as the exponent of the binomial, and decreases by 1 in each succeeding term. 3. The exponent of x in the second term is 1, and increases by 1 in each succeeding term. 4. The coefficient of the first term is 1, and the coefficient of the second term is the exponent of the binomial. 5. If the coefficient of any term be multiplied by the exponent of a in that term, and the result divided by the exponent of x in the term increased by 1, the quotient will be the coefficient of the next following term. 252. If the laws of § 251 be assumed to hold for the expansion of (a+x)", where n is any positive integer, the exponent of a in the first term is n, in the second term n−1, in the third term n-2, in the fourth term n-3, etc. The exponent of x in the second term is 1, in the third term 2, in the fourth term 3, etc. The coefficient of the first term is 1; of the second term n. Multiplying the coefficient of the second term, n, by n-1, the exponent of a in that term, and dividing the result by the exponent of x in the term increased by 1, or 2, we have n(n-1) as the coefficient of the third term; and so on. 1.2 n−1) an-2x2 1.2 Multiplying both members of (1) by a+x, we have Collecting the terms which contain like powers of a and x, we have n+1 an-1x2 Then, (a+x)+1=a"+1+ (n+1)a"x+n It will be observed that this result in equation (2) is of the same form in n+1, that equation (1) is in n, and equation (2) was obtained by multiplying equation (1) by a+x; which proves that, if the laws of § 251 hold for any power of a+ whose exponent is a positive integer, they also hold for a power whose exponent is greater by 1. But the laws have been shown to hold for (a+x)1, and hence they also hold for (a+x)5; and since they hold for (a+x), they also hold for (a+x); and so on. Therefore, the laws hold when the exponent is any positive integer, and equation (1) is proved for every positive integral value of n. Equation (1) is called the Binomial Theorem. In place of the denominators 1.2, 1.2.3, etc., it is usual to write 12, 13, etc. The symbol n, read "factorial-n," signifies the product of the natural numbers from 1 to n, inclusive. The method of proof in § 252 is known as Mathematical Induction. 253. Putting a=1 in equation (1), § 252, we have (1+x)=1+nt n(n−1) x2 + 2 n(n−1)(n−2) x3 +.... 3 254. In expanding expressions by the Binomial Theorem, it is convenient to obtain the exponents and coefficients of the terms by aid of the laws of § 251. 1. Expand (a+x). The exponent of a in the first term is 5, and decreases by 1 in each succeeding term. The exponent of x in the second term is 1, and increases by 1 in each succeeding term. The coefficient of the first term is 1; of the second, 5. Multiplying 5, the coefficient of the second term, by 4, the exponent of a in that term, and dividing the result by the exponent of x increased by 1, or 2, we have 10 as the coefficient of the third term; and so on. Then, (a+x)=a5+5 a1x + 10 a3x2 +10 a2x3 +5 ax1+x3. It will be observed that the coefficients of terms equally distant from the ends of the expansion are equal. Thus the coefficients of the latter half of an expansion may be written out from the first half. If the second term of the binomial is negative, it should be written, negative sign and all, in parentheses before applying the laws; in reducing, care must be taken to apply the principles of § 88. 2. Expand (1-x)o. (1-x)=(1+(-x)]® . =1° +6 · 15 · ( − x) + 15 · 11 · ( − x)2 + 20 • 1 3 · ( − x) 3 + 15 · 12 · ( − x)1+6·1 · ( − x)3+(−x)® =1-6x+15 x2-20 x3 +15 x1 − 6 x3 +x®. If the first term of the binomial is an arithmetical number, it is convenient to write the exponents at first without reduction; the result should afterwards be reduced to its simplest form. If either term of the binomial has a coefficient or exponent other than unity, it should be written in parentheses before applying the laws. |