Any suggestions for an approach to proving that the nth term of the following sequence:
1,2,2,3,3,3,4,4,4,4,5,5,5...
is [ ((2*n)^(1/2) + (1/2)) ], where [x] denotes the floor value of x and ^ denotes to the power of
First of all, I have to say that this is a very good exercise.
Consider the terms marked with , for these numbers, we have
Now, let , where is the smallest number which is not less than and satisfies (in short, lets say property) and is the greatest number which is less than and satisfies property.
Therefore, solving this parabola with, we have . Note that the desired term is the positive one, hence . Thus, the solution is completed when has property.
If does not have property, then we see that , you can easily show that , because of the definitions of , being and the sequezing
Plato's answer is right!