Thus the expansion of the proposed multinomial consists of a series of terms of which that just given may be taken as the general type. ...... It should be observed that q, r, s, t, are always positive integers, but p is not a positive integer unless n be a positive integer. When p is a positive integer, we may, by multiplying both numerator and denominator by p, write the coefficient n (n − 1) (n − 2) in the more symmetrical form In Pq r s t (p + 1) 529. Suppose we require the coefficient of an assigned power of x in the expansion of (a,+ a ̧x+а ̧x2 + ......)", for example, that of x". We have then q, r, s, t, ...... p+q+r+s+t+...... =n. We must find by trial all the positive integral values of which satisfy the first of these equations; then The required coefficient is then the sum of the corresponding values of the expression from the second equation p can be found. When n is a positive integer, then p must be so too, and we may use the more symmetrical form 530. For example, find the coefficient of x2 in the expansion of (1+2x+3x2 + 4x3)*. Begin with the greatest admissible value of s; this is s = 2, with which we have r = 0, q = 1, p = 1. Next try s = 1; with this we may have r = 2, q = 0, p = 1; also we may have r=1, q=2, p=0. Next try s = 0; with this we may have r = = 3, q=1, p=0. These are all the solutions; they are collected in the annexed table. Also a,1, a, 2, a,= 3, a,= 4. = Thus the required coeffi that is, 384 + 432 + 576 +216; that is, 1608. Again; find the coefficient of x3 in the expansion of 3 0 0 p+q+r+s+ = ...... All the solutions are given in the annexed table, and the required coefficient is 312 23; the proposed expression is {(1 − x)-2, that is, (1 − x)-'. And (1 − x) ̄1 = 1 + x + x2 +∞3 + ...... ; T. A. 21 thus we see that the coefficient of x3 ought to be 1; and the student may exercise himself by applying the multinomial theorem to find the coefficients of other powers of x, as, for example, x1. EXAMPLES OF THE MULTINOMIAL THEOREM. Find the coefficients of the specified powers of x in the following expansions : 1. x* in (1 + x + x3)3. 2. x12 in (1+α ̧x+a ̧x3 + ɑ¿x3)3. 3. x3 in (1 - 2x + 3x3 — 4x3)*. 4. x11 in (1 + x + x2 + x3 + x2 + x3)3. 5. x in (2-3x-4x2). 6. x3 in (1−x + 2x2)12. ༧॰) ཡ༔ 21. 2 in (1-x2 + x3 — x3)*. 22. x2 in (1 + ax + bx2)−§. 23. x3 in (1+a ̧x +α ̧x2 +α ̧æ3 +......)". 24. Find the coefficient of abc3 in (a+b+c)3. 25. Find the coefficient of a2b3c3 in (a − b − c)". 28. Write down those terms in the expansion of (a+b+c)" which involve powers of b and c as high as the third power inclusive. 29. Write down all the terms in the expansion of which contain d"¬3. (a+b+c+d)" 30. Find the greatest coefficient in the expansion of (a+b+c+d)1o. 31. The greatest coefficient in the expansion of is the quotient, and r the remainder when n is divided 32. Shew that the coefficient of x2+1 in the expansion of (α + a ̧x + α ̧x2 + ......)2 a positive integer, shew that (1) the coefficients of the terms equi distant from the beginning and the end are equal; (2) the coefficient of the middle term, or of the two middle terms, according as nr is even or odd, is greater than any other coefficient; (3) the coefficients continually increase from the first up to the greatest. 531. XXXVIII. LOGARITHMS. Suppose an, then x is called the logarithm of n to the base a; thus the logarithm of a number to a given base is the index of the power to which the base must be raised to be equal to the number. The logarithm of n to the base a is written log.n; thus logan = x expresses the same relation as a* = n. 532. For example, 381; thus 4 is the logarithm of 81 to the base 3. = ...... ...... If we wish to find the logarithms of the numbers 1, 2, 3, to a given base 10, for example, we have to solve a series of equations 101, 102, 10′′ 3, . We shall see in the next chapter that this can be done approximately, that is, for example, although we cannot find such a value of x as will make 10* = 2 exactly, yet we can find such a value of x as will make 10 differ from 2 by as small a quantity as we please. We shall now prove some of the properties of logarithms. 533. The logarithm of 1 is 0 whatever the base may be. For a 1 when x=0. |