# Estimate the sum to the nearest whole number

#### greg1313

Estimate the sum to the nearest whole number without the use of a calculator/computer.

$$\displaystyle \displaystyle\sum_{n = 625}^{9999} \sqrt[4]{n + 0.5}$$

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#### romsek

MHF Helper
amazing

eagerly awaiting the simple explanation

#### Debsta

MHF Helper
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!

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#### Debsta

MHF Helper
A (much) better estimate:

$$\displaystyle \int_{625}^{9999} (n+0.5)^\frac{1}{4} dn$$

$$\displaystyle \approx \frac{4}{5} (10000^ \frac{5}{4} - 625^\frac{5}{4})$$

$$\displaystyle = \frac{4}{5} (10^5 - 5^5)$$

$$\displaystyle = 77 500$$

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