Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Quick Sum

According to my calculator, I see that

$$

\frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}} + \text{and so on up to}+\frac{1}{ \sqrt {15}+ \sqrt{16}} = 3.0000000000

$$

How could I prove that the answer is, indeed, exactly 3?

Or search by topic

Age 16 to 18

ShortChallenge Level

- Problem
- Getting Started
- Solutions

According to my calculator, I see that

$$

\frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}} + \text{and so on up to}+\frac{1}{ \sqrt {15}+ \sqrt{16}} = 3.0000000000

$$

How could I prove that the answer is, indeed, exactly 3?

Did you know ... ?

Mathematicians often use calculating aids to help form an opinion concerning the likely numerical answer to a problem involving irrational numbers, but will always seek a full proof which does not rely on the calculator to complete the problem.

Mathematicians often use calculating aids to help form an opinion concerning the likely numerical answer to a problem involving irrational numbers, but will always seek a full proof which does not rely on the calculator to complete the problem.