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work out the value of 1/square root of 1 + square root of 2 + 1/square root of 2 + square root of 3 + . . . + 1/square root of 63 + square root of 64

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Final answer:

The sum of the given series of radicals and rational expressions is equal to the square of the number of terms, n². The pattern in the series is due to it being a telescoping series where terms cancel each other when simplified. For the sum up to 1/(√63+√64), it would be 64 since 64 is the square of the 8th term.

Step-by-step explanation:

The student's question involves calculating the sum of a series that combines radicals and rational expressions. We are given that for n terms the expression equals n². This series is a telescoping series, where each subsequent fraction after the first has a numerator that will cancel out the denominator of the previous fraction.

The pattern suggests that when adding fractions such as 1/(√1+√2), 1/(√2+√3), ..., 1/(√63+√64), we need to rationalize the denominators to reveal a form that collapses when summed. The final expression simplifies to the square of the number of terms.

So, for example, if we have 8 terms, the sum would be 8² or 64.

In conclusion, the value of this series for any integer n can be found using the equation 1/(√1+√2) + 1/(√2+√3) + ... + 1/(√(n-1)+√(n)) = n².

Therefore, for the given expression up to √64, which is the 8th term, the sum would be 8² = 64.

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