Estimate the sum to the nearest whole number without the use of a calculator/computer.
$\displaystyle \displaystyle\sum_{n = 625}^{9999} \sqrt[4]{n + 0.5}$
First thoughts. Here's one estimate:
If $\displaystyle n=5$ then $\displaystyle \sqrt[4]{n+0.5} \approx \sqrt[4]{625} = 5 $
If $\displaystyle n=9999$ then $\displaystyle \sqrt[4]{n+0.5} \approx \sqrt[4]{10000} = 10 $
So the first number in the sum is approx 5 and the last number is approx 10. As n increases, the value of $\displaystyle \sqrt[4]{n+0.5}$ also increases.
There are 9999 - 625 +1 = 9375 terms in the sum.
Let's assume it is an AP (it isn't, but this will give us an approximation of the sum since the range of values 5 to 10 is quite small and there are a lot of terms).
So, using the sum of an AP formula, Sum $\displaystyle \approx \frac{9375}{2}(5+10) \approx70313$
(This gives the same result as if we say each term is approx 7.5 (ie halfway between 5 and 10)).
So an estimate is 70313.
Of course there would be other ways and this probably isn't the best!
A (much) better estimate:
$\displaystyle \int_{625}^{9999} (n+0.5)^\frac{1}{4} dn$
$\displaystyle \approx \frac{4}{5} (10000^ \frac{5}{4} - 625^\frac{5}{4})$
$\displaystyle = \frac{4}{5} (10^5 - 5^5)$
$\displaystyle = 77 500 $