BINOMIAL THEOREM. 345
16. Find the value of (x - 3)4 + (x+ 3)4. 17. Expand (1-x+ 1)5 - (1- x2 - 1)5. 18. Find the coefficient of x12 in (x3 + 2x)10. 19. Find the coefficient of x in (x2-a 2x) 20. Find the term independent of x in (2x2-1 x)12 21. Find the coefficient of x-20 in (x2 3 - 2 x3)15.
415. Equal Coefficients. In the expansion of (1-x)n the coefficients of terms equidistant from the beginning and end are equal.
The coefficient of the (r+1)th term from the beginning is nCr.
The (r+1)th term from the end has n+1-(r+1), or n-r terms before it; therefore counting from the beginning it is the (n-r+1)th term, and its coefficient is nCn-r which has been shown to be equal to nCr [Art. 395]. Hence the proposition follows.
416. Greatest Coefficient. To find the greatest coefficient in the expansion of (1+x)n.
The coefficient of the general term of (1+x)n is nCr; and we have only to find for what value of + this is greatest.
By Art. 401, when n is even, the greatest coefficient is nCn 2 when n is odd, it is nCn-1 2, or nCn+1 2 these coefficients being equal.
417. Greatest Term. To find the greatest term in the expansion of (a+b)n.
We have (a+b)n = an (1+b a)n;
therefore, since an multiplies every term in (1+b a)n , it will be sufficient to find the greatest term in this latter expansion.
