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
let $\displaystyle A_n$ be the n-th term of the sequence. it's easy to see that your sequence can be written as: $\displaystyle A_{\frac{m(m-1)}{2} + k}=m, \ \ m \geq 1, \ 1 \leq k \leq m.$ now let:
$\displaystyle \frac{m(m-1)}{2}+k=n, $ where $\displaystyle m \geq 1$ and $\displaystyle 1 \leq k \leq m.$ so $\displaystyle A_n=m,$ and: $\displaystyle 8n=4m^2-4m+8k=(2m-1)^2+8k-1.$ thus : $\displaystyle 8n > (2m-1)^2. \ \ \ (1)$
also we have: $\displaystyle 8n=(2m+1)^2-8(m-k)-1.$ thus: $\displaystyle 8n < (2m+1)^2. \ \ \ \ (2)$
taking square root in (1) and (2) will give us: $\displaystyle m < \sqrt{2n} + \frac{1}{2} < m+1,$ which means: $\displaystyle \left \lfloor \sqrt{2n} + \frac{1}{2} \right \rfloor=m=A_n.$
Here is another way to generate that sequence: $\displaystyle a_n = \left\lceil {\frac{{ - 1 + \sqrt {8n + 1} }}{2}} \right\rceil $.
That is of course using the ceiling function.
Notice the sequence changes on the triangular numbers, 1,3,6,10,…; $\displaystyle \frac{{n(n + 1)}}{2}$.
First of all, I have to say that this is a very good exercise.
$\displaystyle
\begin{array}{cccccccccccc}
n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & \cdots \\
a_{n} & 1 & 2 & 2 & 3 & 3 & 3 & 4 & 4 & 4 & 4 & \cdots\\
& * & & * & & & * & & & & * &
\end{array}
$
Consider the terms marked with $\displaystyle *$, for these numbers, we have $\displaystyle n=\sum\limits_{i=1}^{a_{n}}i=\frac{a_{n}(a_{n}+1)} {2}.$
Now, let $\displaystyle a_{n_{0}}\leq a_{n}\leq a_{n^{0}}$, where $\displaystyle n^{0}\geq n$ is the smallest number which is not less than $\displaystyle n$ and satisfies $\displaystyle n^{0}=\frac{a_{n^{0}}(a_{n^{0}}+1)}{2}$ (in short, lets say $\displaystyle *$ property) and $\displaystyle n_{0}\leq n$ is the greatest number which is less than $\displaystyle n$ and satisfies $\displaystyle *$ property.
Therefore, solving this parabola with, we have $\displaystyle a_{n^{0}}=\frac{-1\pm\sqrt{1+8n^{0}}}{2}$. Note that the desired term is the positive one, hence $\displaystyle a_{n^{0}}=\frac{-1+\sqrt{1+8n^{0}}}{2}$. Thus, the solution is completed when $\displaystyle n$ has $\displaystyle *$ property.
If $\displaystyle n$ does not have $\displaystyle *$ property, then we see that $\displaystyle n_{0}<n<n^{0}$, you can easily show that $\displaystyle a_{n}=\left\lceil\frac{-1+\sqrt{1+8n}}{2}\right\rceil$, because of the definitions of $\displaystyle n_{0},n^{0}$, being $\displaystyle a_{n_{0}}+1=a_{n^{0}}=a_{n}$ and the sequezing $\displaystyle n_{0}=\frac{a_{n_{0}}(a_{n_{0}}+1)}{2}\leq n\leq\frac{a_{n^{0}}(a_{n^{0}}+1)}{2}=n^{0}.$
$\displaystyle \therefore$ Plato's answer is right!