UPSC CDS 1 2022 Math Paper

Given:
‌

1

1+√2

+‌

1

√2+√3

+‌

1

√3+√4

+...+‌

1

√2020+√2021

Concept used:
(a+b)(a−b)=a2−b2
Calculation:
Consider the first term only, i.e. ‌

1

1+√2

Rationalize the denominator
⇒‌

1

1+√2

×‌

√2−1

√2−1

⇒‌

√2−1

(√2)2−(1)2

⇒√‌2−1
After doing a similar operation in other terms, we will get
√2−1+√3−√2+√4−√3......+√2021−√2020
⇒√2021−1
∴ Required value of the expression is √2021−1