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Telescoping sums of reciprocal surd terms by rationalising

This type of question, asked 3 times: CDS 2 2020, CDS 1 2022, CDS 2 2024. All 4 questions below, oldest first.

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  1. Question 1

    CDS 2 2020

    12
    What is the value of 11+2+12+3+13+4+…+199+100\frac{1}{1+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{4}} + \ldots + \frac{1}{\sqrt{99}+\sqrt{100}} ?

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  2. Question 2

    CDS 1 2022

    8
    What is the value of the following? 154+45+165+56+176+67+187+78+198+89\frac{1}{5\sqrt{4} + 4\sqrt{5}} + \frac{1}{6\sqrt{5} + 5\sqrt{6}} + \frac{1}{7\sqrt{6} + 6\sqrt{7}} + \frac{1}{8\sqrt{7} + 7\sqrt{8}} + \frac{1}{9\sqrt{8} + 8\sqrt{9}}

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  3. Question 3

    CDS 1 2022

    24
    What is 11+2+12+3+13+4+⋯+12020+2021\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \cdots + \frac{1}{\sqrt{2020} + \sqrt{2021}} equal to?

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  4. Question 4

    CDS 2 2024

    8
    What is 110+9+111+10+112+11+…+1196+195\frac{1}{\sqrt{10} + \sqrt{9}} + \frac{1}{\sqrt{11} + \sqrt{10}} + \frac{1}{\sqrt{12} + \sqrt{11}} + \ldots + \frac{1}{\sqrt{196} + \sqrt{195}} equal to ?

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