Telescoping series: split each term and cancel to find the sum

This type of question, asked 3 times: CDS 2 2016, CDS 1 2021, CDS 1 2024. All 3 questions below, oldest first.

Try each one first, then open its answer.

Question 1
CDS 2 2016
25 $\sqrt{1 + \frac{1}{1^2} + \frac{1}{2^2}} + \sqrt{1 + \frac{1}{2^2} + \frac{1}{3^2}} + \ldots + \sqrt{1 + \frac{1}{2007^2} + \frac{1}{2008^2}}$
Tap an option to check your answer.
Question 2
CDS 1 2021
30 $\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \cdots + \frac{1}{n(n + 1)} = \frac{99}{100}$
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Question 3
CDS 1 2024
35 $\frac{3}{1^2 \times 2^2} + \frac{5}{2^2 \times 3^2} + \frac{7}{3^2 \times 4^2} + \ldots$
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Question 1