Answer 1

`a^2 +b^2`

(1 + i)(1 + 2i)(1 + 3i) ......(1 + ni) = a + ib

Taking modulus on both the sides, we get:

`|(1+i)(1+2i) (1+3i).............. (1+ni)| = |a+ib|`

`|(1+i)(1+2i)(1+3i)..............(1+ni)|`can be written as `|(1+i)|  |(1+2i)|  |(1+3i)|........|(1+ ni)|`

\[\sqrt{1^2 + 1^2} \times \sqrt{1^2 + 2^2} \times \sqrt{1^2 + 3^2} \times . . . \times \sqrt{1 + n^2} = \sqrt{a^2 + b^2}\]

\[\Rightarrow \sqrt{2} \times \sqrt{5} \times \sqrt{10} \times . . . \times \sqrt{1 + n^2} = \sqrt{a^2 + b^2}\]

Squaring on both the sides, we get:

\[2 \times 5 \times 10 \times . . . \times (1 + n^2 ) = a^2 + b^2\]