CHAPTER XX. THE BINOMIAL THEOREM. 248. We have already [Art. 67] proved that the continued product of any number of algebraical expressions is the sum of all the partial products which can be obtained by multiplying any term of the first, any term of the second, any term of the third, &c. 249. Binomial Theorem. Suppose that we have n factors each of which is a + b. If we take a letter from each of the factors of (a + b) (a + b) (a+b)............ and multiply them all together, we shall obtain a term of the continued product; and if we do this in every possible way we shall obtain all the terms of the continued product. [Art. 67.] Now we can take the letter a every time, and this can be done in only one way; hence a" is a term of the product. The letter b can be taken once, and a the remaining (n-1) times, and the number of ways in which one b can be taken is the number of ways of taking 1 out of n things, so that the number is C: hence we have n Again, the letter b can be taken twice, and a the remaining (n-2) times, and the number of ways in which two b's can be taken is the number of ways of taking 2 out of n things, so that the number is C2: hence we have n 2 And, in general, b can be taken r times (where r is any positive integer not greater than n) and a the remaining nr times, and the number of ways in which r b's can be taken is the number of ways of taking r out of n things, so that the number is C: hence we have Thus n "C. a"-"b". (a + b) (a + b) (a+b).............. to n factors 2 =a” + „C1 . a” ̄1b + „C2. an¬2b2 + ...... +„C.... a” ̃ ̃b” + ... 1 n 2 the last term being „Ca"-"b", i.e. b”. Hence, when n is any positive integer, we have The above formula is called the Binomial Theorem. If we substitute the known values [see Art. 240] of C,,, C, C,... in the series on the right we obtain the form in which the theorem is usually given, namely n 1' n 2 n The series on the right is called the expansion of (a + b)". 250. Proof by Induction. The Binomial Theorem may also be proved by induction, as follows. We have to prove that, when n is any positive integer, 2-2 (a+b)" = a” + „C1 aˆ-1 b + „C2 a"¬2 b2 +...+„C, aˆ ̃ b* +...+b”. n 1 n 2 n Now if we assume that the theorem is true when the index is n, and multiply by another factor a+b, we have, when like terms of the product are collected, (a + b)n+1 = an+1 + (1 + n C1) a′′b + („C1 + „C2) an−1b2 1 n Thus if the theorem be true for any value of n, it will be true for the next greater value. = 2, Now the theorem is obviously true when n=1. Hence it must be true when n = 2; and being true when n = it must be true when n = 3; and so on indefinitely. The theorem is therefore true for all positive integral values of n. Put 2x for a, and -y for b in the general formula: then 3.2 (2x − y)3 = (2x)3 +3 (2x)2 ( − y) + (2x) ( − y)2+ = 8x3 – 12x2y + 6xy2 — y3. Ex. 3. Expand (a - b)". Change the sign of b in the general formula; then we have 251. General term. By the preceding articles we see that any term of the expansion of (a+b)" by the Binomial Theorem will be found by giving a suitable value to r in On this account, the above is called the general term of the series. It should be noticed that the term is the (r+1)th term from the beginning. [See Note Art. 240.] 252. Coefficients of terms equidistant respectively from the beginning and the end are equal. In the expansion of (a+b)" by the Binomial Theorem, the (r+1)th term from the beginning and the (r+1)th term from the end are respectively But „C,.a"-b" and "Cn-r.a" br. nCr=nCn-r [Art. 241.] Hence, in the expansion of (a + b)", the coefficients of any two terms equidistant respectively from the beginning and the end are equal. This result follows, however, at once from the fact that (a+b)", and therefore also its expansion, would be unaltered by an interchange of the letters a and b; and hence the coefficient of a b must be equal to the coefficient of b” ̄* a”. n-x 253. If, in the formula of Art. 249, we put a = 1 and b=x, we have This is the most simple form of the Binomial Theorem, and the one which is generally employed. The above form includes all possible cases: if, for example, we want to find (a+b)" by means of it, we have 254. α = a" 1+n + b n (n − 1)/b 1.2 α Greatest term of a binomial expansion. In the expansion of (1+x)", the (r+1)th term is formed from the rth by multiplying by n -r+1 x. r |