| Expansion |
| The binomial expansion of (x-1/2+x-1/3)n, where n is a positive integer, contains a term in the form c*x-17/6, where c is a constant, for three different values of n. Find the value(s) of c. |
Problem Moderated by: Douglas |
| Problem Solution |
Define tk (disregarding the coefficient) in the binomial expansion of (x-1/2+x-1/3)n as follows:
tk=(x-1/2)n-k+1(x-1/3)k-1=xk-3n-1/6=x-17/6
So k-3n-1=-17, and k=3n-16. We find that the only lattice points satisfying both k=3n-16 and k<=n+1 are as follows:
k=2 and n=6
k=5 and n=7
k=8 and n=8
The binomial coefficient of these terms is the number of combinations possible with n objects taken k-1 at a time - or, respectively, 6, 35, and 8.
We can also solve this problem using arbitrary exponents a and b, provided that a and b are whole numbers (non-negative integers):
(x-1/2)a(x-1/3)b=x-3a-2b/6=x-17/6
3a+2b=17
17-3a must be an even number, so a must be odd. Therefore, we can have a=1 and b=7, a=3 and b=4, or a=5 and b=1, and the binomial coefficient will be the number of ways that a+b be objects can be taken a at a time - 8, 35, and 6. |
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