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!