Now add (1) and (2); the irrational terms on the right disappear, and we have But F and F are proper fractions: we must therefore have A similar result holds for (a+b)" if a is the integer next greater than b, so that a√b is a proper fraction. (4) Required the sum of the coefficients of the first r+1 terms of the expansion of (1-x)". We have Therefore (1 − x)−("+1) is equal to the product of the two series. Now if we multiply the series together, we see that the coefficient of x in the product is this must therefore be equal to the coefficient of x in the expansion of (1-x)-(+); that is, to (5) The Binomial Theorem may be applied in the manner just shewn to establish numerous algebraical identities; we will give one more example. $ (n, 0) 4 (n, r) − 4 (n, 1) 4 (n − 1, r − 1) + † (n, 2) o (n − 2, r − 2) The expression here given is the expansion of n (n − 1) (n − 2) ... (n − r + 1) (1-1), EXAMPLES OF THE BINOMIAL THEOREM. Expand each of the following twelve expressions to four terms: Find the (r+1)th term in the expansion of the following seven expressions: 1 13. (1−x)~'. 14. (1-x)2. 15. (1−pœ). 16. √(1+x)' 17. (1−x2)-3⁄4. 18. (1-2x). 1 19. /(1-x)* Calculate the following four roots approximately: 20. (24). 21. /(999). 22.5/(31). 23. 5/(99000). 24. If x be small compared with unity, shew that 25. Shew that the number of combinations of n things when taken in ones, threes, fives, ...... exceeds the number when taken in twos, fours, sixes, ...... by unity. 26. Shew that the number of homogeneous products of n things of n dimensions is Find the greatest term in the following four expansions : 31. Find the greatest term in the expansion of where n is a positive integer. 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 36. What is the coefficient of a" in the expansion of (1+x)"} (1-x) 4 37. Expand (+)* in ascending powers of z. Write down 38. Prove that the nth coefficient in the expansion of (1 − x)-" is always the double of the (n-1)th. 39. Shew that if t denote the middle term in the expansion 41. Find the sum of the squares of the coefficients in 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 44. Find the coefficient of x" in (1 + 2x + 3x2 + 4x3 + ...... ad inf.)". 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,+a,x+ax2 + α ̧3+......)". The number of terms in the series a,, a,, ɑ ̧, may be any whatever, and ʼn may be positive or negative, integral or fractional. ...... Put b, for ax + а ̧x2 +α ̧Ã3 + ......, then we have to expand (a+b)"; the general term of the expansion is 2 μ being a positive integer. Put b, for ax2+ax3+......, then b,”=(a,x+b2)"; since μ is a positive integer the general term of the expansion of (ax+b),μ may be denoted either by we will adopt the latter form as more convenient for our purpose. Combining this with the former result, we see that the general term of the proposed expansion may be written bu = Again, put b, for ax3+ax+......, then bμ-- (ax2+b.)”—, and the general term of the expansion of this will be 3 Hence the general term of the proposed expansion may be written 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 |