UPSC CDS 1 2022 Math Paper

Given:
‌

1

5√4+4√5

+‌

1

6√5+5√6

+‌

1

7√6+6√7

+‌

1

8√7+7√8

+‌

1

9√8+8√9

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

1

5√4+4√5

Rationalize the denominator
⇒‌

1

5√4+4√5

×‌

5√4−4√5

5√4−4√5

⇒‌

5√4−4√5

(5√4)2−(4√5)2

⇒‌

5√4−4√5

100−80

=‌

5√4

20

−‌

4√5

20

⇒‌

1

√4

−‌

1

√5

After doing a similar operation in other terms, we will get
‌

1

√4

−‌

1

√5

+‌

1

√5

−‌

1

√6

+‌

1

√6

−‌

1

√7

+‌

1

√7

−‌

1

√8

+‌

1

√8

−‌

1

√9

⇒‌

1

√4

−‌

1

√9

⇒‌

1

2

−‌

1

3

⇒‌

3−2

6

⇒‌

1

6

∴ Required value of the expression is ‌

1

6