Let z = (1 + i) (1 + 2i) (1 + 3i) .... (l + ni), where i = $$\sqrt{-1}$$. If $$|z|^{2}$$ = 44200, then n is equal to __

We need to find n such that $$|z|^2 = 44200$$ where $$z = (1+i)(1+2i)(1+3i)\cdots(1+ni)$$.

By the property of moduli, $$|z|^2 = |1+i|^2 \cdot |1+2i|^2 \cdot |1+3i|^2 \cdots |1+ni|^2$$ which simplifies to $$= (1+1)(1+4)(1+9)\cdots(1+n^2) = 2 \cdot 5 \cdot 10 \cdots (1+n^2).$$

We then compute this product for successive values of n. For n = 1 it is 2; for n = 2 it becomes 2 × 5 = 10; for n = 3 it yields 10 × 10 = 100; for n = 4 it gives 100 × 17 = 1700; and for n = 5 it results in 1700 × 26 = 44200.

Since we require $$|z|^2 = 44200$$, we conclude that n = 5.