Problem:
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers x,y, and r with ∣x∣>∣y∣,
(x+y)r=xr+rxr−1y+2!r(r−1)xr−2y2+3!r(r−1)(r−2)xr−3y3+…
What are the first three digits to the right of the decimal point in the decimal representation of (102002+1)10/7?
Solution:
Apply the Binomial Expansion to obtain
(102002+1)10/7=102860+710⋅103⋅286+2710⋅73⋅10−4⋅286+…
Thus, only the second term affects the requested digits. Since 1/7=.142857 and 6 is a divisor of 3⋅286, conclude that
710⋅103⋅286=1428571…571.428571
so the answer is 428.
The problems on this page are the property of the MAA's American Mathematics Competitions