EXAMPLES OF THE BINOMIAL THEOREM.

Expand each of the following expressions to four terms:

Find the (r+ 1) term in the expansions of the following

24. If x be small compared with unity, shew that

25. Shew that the number of combinations of n things taken

in ones, threes, fives,

......

twos, fours,...... by unity.

exceeds the number when taken by

26. Shew that the number of homogeneous products of n things of n dimensions is

32. Find the number of terms in the expansion of

(a + b + c + d)1o.

33. Find the first term with a negative coefficient in the

11

expansion of (1 + x)3.

34. If p be greater than n, the coefficient of x in the expanx2 p (p-1) (p2 - 23)......

sion of

(1 − x)2n

is

35. The coefficient of x2 in

3n-1

{p2 - (n − 1)"}

36. What is the coefficient of x" in

(1 + x)2

(1-x)*

?

38. Prove that the n coefficient of the expansion of (1-x)" is always the double of the (n - 1)th.

39. Shew that if t, denote the middle term of (1 + x)3, then

41. Find the sum of the squares of the coefficients of the expansion of (1 + x)", where n is a positive integer.

43. Prove that the coefficient of x in the expansion of

is equal to the coefficient of x" in the expansion of

1

(1-x)+1

44. Find the coefficient of x in (1 + 2x + 3x2 + 4x3 +

XXXVII. THE MULTINOMIAL THEOREM.

528. We have in the preceding chapter given some examples of the expansion of a multinomial; we now proceed to consider this point more fully. We propose to find an expression for the general term in the expansion of (a +а ̧x+α ̧x2 +a ̧ñ3 + ..............)". The number of terms in the series a, a, α may be any whatever, and n may be positive or negative, integral or fractional.

......

Put b, for ax + а ̧x2 + a ̧ï3 +

.......

then we have to expand

(a+b)"; the general term of the expansion is

b," = (a,x+b)"; since μ is a positive integer the general term of the expansion of (ax + b)" may be denoted by

Combining this with the former result, we see that the general term of the proposed expansion may be written

= 2

Again put b, for ax+a+...... then b" (a ̧x2 + b)μ~", and the general term of the expansion of this will be

Hence the general term of the proposed expansion may be

Proceeding in this way we shall obtain for the required. general term

If we suppose n-μ=p, we may write the general term in

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

529. Suppose we require the coefficient of an assigned power of x in the expansion of (a + а ̧x + α ̧x2 + ......)", for example, that of x. We have then

We must find by trial all the positive integral values of q, r, s, t, ...... which satisfy the first of these equations; then from the second equation p can be found. The required coefficient is then the sum of the corresponding values of the expression

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 x in the expansion of (1 + 2x + 3x2 + 4x3)*.