Math Problem Statement
The value of $\small \frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{8}}+\frac{1}{\sqrt{8}+\sqrt{9}}$ is:
Solution
We are asked to find the value of the following sum:
Step 1: Rationalizing the denominators
Each term in the sum has the form , where is an integer. To simplify this, we rationalize the denominator by multiplying both the numerator and the denominator by .
For example, for the first term , we multiply the numerator and denominator by :
Similarly, for the second term , we multiply by :
Step 2: Applying this to all terms
We can apply this rationalization to all terms in the sum. The general term will look like this:
So, each term in the sum becomes:
Thus, the sum becomes:
Step 3: Simplifying the sum
Notice that this is a telescoping series. Most terms cancel out with adjacent terms:
Thus, the value of the sum is:
Would you like further explanation or any other details? Here are some related questions:
- How does the telescoping property help simplify sums like this?
- What happens if we have a different number of terms in the sum?
- Can this method be applied to sums with different types of radicals?
- How do we rationalize denominators in general when dealing with square roots?
- What other types of sums can be simplified using similar techniques?
Tip: Always look for cancellation patterns (like telescoping) when summing terms that have similar structures.
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Math Problem Analysis
Mathematical Concepts
Algebra
Rationalization of Denominators
Telescoping Series
Radicals
Formulas
Rationalizing the denominator: \frac{1}{\sqrt{n} + \sqrt{n+1}} = \sqrt{n+1} - \sqrt{n}
Sum of a telescoping series: \sum_{n} (a_{n} - a_{n+1}) = a_1 - a_n
Theorems
Rationalization of square roots
Telescoping Series
Suitable Grade Level
Grades 9-12
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