2. What is the 10th power of a + x? Ans. a1o + 10 a3 x + 45 a3 x2 + 120 a2 x3 + 210 a¤ x2 + ··· 252 a3 x + 210 a1 xo + 120 a3 x2 + 45 a2x2 + 10 a x + x1o. 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 a c = b. The 2d power of b+d is b2 + 2 b d +ď. Putting 2 ac the value of b into this, instead of b, observing that b2 4 a' c2, and it becomes 4a2 c2+4acd+d2. 6. What is the 3d power of 3 c2+2bd? Make a 3 c2 and x — 2 b d. The 3d power of a +x is a3 + 3 a2 x + 3 a x2 + x3. Put into this the values of a and a and it becomes 27c+54 cbd + 36 c2 b2 d2 + 8b3 ď3, which is the 3d power of 3 c2+2b d. 7. What is the 3d power of a b? Make xb, then having found the 3d power of a + x put-b. in the place of x 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 +. 8. What is the 7th power of m — n ? 11. What is the 3d power of a+b+c? · The 3d power of a +m is a3 + 3 a2 m + 3 a m3 + m2 m3 = b3 + 3 b2 c + 3 b c2 + c3. Substituting these values of m, the third power of a+b+c will be a3+3a2b+3a2c+3 a b2+ 6 a b c +3 ac2+b3+3b2 c+3bc2+c3. 12. What is the 3d power of a-b+c? Make a b = m, raise m +c to the 3d power, and then sub stitute the value of . Ans. a3-3 ab+3a2 c + 3 a b2 6 a b c +3 a c2 — b3.... +362 c-3 b c2 + c3 ; 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. a2+2ab+b2+2ac+2bc-2ad-2bd+c2. 14. What is the 3d power of 2 a 15. What is the 7th power of 3 a' — 2 a2 d? -2cd+d. 16. What is the 4th power of 7 b2 + 2 c — d3 ? 17. What is the 13th power of a3 — 2 b2 ? 20. What is the 3d power of a-b-2 cď? α 21. What is the 5th power of 7 a b3 — 10 a3 c2? 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 The sum of (n 1) terms of the series of the second order is n (n 1 X 2 The sum of (n-2) terms of the series of the third order is It may 1 X 2 X 3 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 The coefficient of the third term is multiplied by, or 1 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. Multiply 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 x, 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 × a1x "+1 which is equal to zero. 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: The first of these is the coefficient of the second term; the coefficient of the second multiplied by g forms the coefficient of the third term, &c. Now 35 multiplied by = 1 will not be altered; hence two successive coefficients will be alike. 21 multiplied by produced 35; so 35 multiplied by 3 must reproduce 21. In this way all the terms will be reproduced; for the last half of the fractions are the first half inverted. This demonstration might be made more general, but it is not necessary. XLVI. Progression by Difference, or Arithmetical Progression. A series of numbers increasing or decreasing by a constant difference, is called a progression by difference, and sometimes an arithmetical progression. The first of the two following series is an example of an increasing, and the second of a decreasing, progression by dif erence. 23. ... 20. 20.... 5, 8, 11, 14, 17, 20, 50, 45, 40, 35, 30, 25, It is easy to find any term in the series without calculating the intermediate terms, if we know the first term, the common difference, and the number of that terin in the series recko ď from the first. Let a be the first term, r the common difference, and n the number of terms. The series is a, a +r, a + 2 r, a + 3 r....a + (n − 2) r, a + (n − 1) r. The points.. ..... are used to show that some terms are left out of the expression, as it is impossible to express the whole until a particular value is given to n. Let / be the term required, then l= a + (n − 1) r. |