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сумму:

(1)/(1+sqrt2)+(1)/(sqrt2+sqrt3)+dots+(1)/(sqrt2006+sqrt2007)

сумму: $\newline$ $\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\ldots+\frac{1}{\sqrt{2006}+\sqrt{2007}}$

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Evaluate integers raised to rational exponents

Full solution

Q. сумму:

\newline

\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\ldots+\frac{1}{\sqrt{2006}+\sqrt{2007}}

Multiply by Conjugate: We can simplify each term by multiplying the numerator and denominator by the conjugate of the denominator. $\newline$ For the first term: $(\frac{1}{1+\sqrt{2}}) \cdot (\frac{1-\sqrt{2}}{1-\sqrt{2}}) = \frac{1-\sqrt{2}}{1-2}$ .
Simplify First Term: Simplify the first term: $(1-\sqrt{2})/(-1) = \sqrt{2}-1$ .
Apply to Second Term: Apply the same process to the second term: $(1)/(\sqrt{2}+\sqrt{3}) \times (\sqrt{3}-\sqrt{2})/(\sqrt{3}-\sqrt{2}) = (\sqrt{3}-\sqrt{2})/(2-3)$ .
Simplify Second Term: Simplify the second term: $(\sqrt{3}-\sqrt{2})/(-1) = \sqrt{2}-\sqrt{3}$ .
Identify Pattern: Notice a pattern: each term simplifies to the difference of two consecutive square roots with alternating signs.
Write Out More Terms: Write out a few more terms to see the pattern: $(\sqrt{2}-1) + (\sqrt{2}-\sqrt{3}) + (\sqrt{3}-\sqrt{4}) + \ldots + (\sqrt{2006}-\sqrt{2007})$ .
Cancel Out Most Terms: Observe that most terms cancel out, leaving only the first and last parts of the sequence.
Simplify Sum: The sum simplifies to: $(\sqrt{2}-1) - (\sqrt{2007})$ .
Final Answer: The final answer is: $-1 - \sqrt{2007} + \sqrt{2}.$

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