Question

1:√1+√2 + 1:√2+√3...1:√2017+√2018

Answer 1

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Answer 2

To simplify the given expression, let's rewrite it in a clearer format:

1/√1 + 1/√2 + 1/√2 + 1/√3 + ... + 1/√2017 + 1/√2018

Now, notice that there are repeating terms with some of the square root values. Specifically, the terms with 1/√2 appear twice, the terms with 1/√3 appear twice, and so on until the terms with 1/√2017 appear twice. The only term without a repetition is 1/√1.

Let's group the terms accordingly:

(1/√1) + (1/√2 + 1/√2) + (1/√3 + 1/√3) + ... + (1/√2017 + 1/√2017) + (1/√2018)

Simplifying:

1 + 2(1/√2) + 2(1/√3) + ... + 2(1/√2017) + 1/√2018

Now, let's notice that the expression inside the parentheses is twice the sum of the reciprocals of square roots from 2 to 2017:

2(1/√2 + 1/√3 + ... + 1/√2017)

Now, we can factor out the common term of 2 and rewrite the sum as follows:

2 * (1/√2 + 1/√3 + ... + 1/√2017) = 2 * (√2 - 1 + √3 - √2 + √4 - √3 + ... + √2017 - √2016)

Notice that most of the terms in the expansion will cancel out, leaving only the first and the last term:

= 2 * (√2017 - 1)

Finally, add the remaining term:

= 2√2017 - 2

So, the simplified form of the given expression is 2√2017 - 2.