Let z = ( 1 + i ) ( 1 + 2 i ) ( 1 + 3 i ) ... ( 1 + n i ) , where i = - 1 . If | z | 2 = 44200 , then n is equal to _________. [2026]

Question

Let $z = (1 + i) (1 + 2 i) (1 + 3 i)$ ... $(1 + n i),$ where $i = \sqrt{- 1}$ . If $| z |^{2} = 44200$ , then $n$ is equal to _________. [2026]

Smart Achievers · Accepted Answer

(5)

$| z |^{2}$ $= 2^{3} \cdot 5^{2} \cdot 13 \cdot 17$

$\prod_{r = 1}^{n} (1 + r^{2}) = 2^{3} \cdot 5^{2} \cdot 13 \cdot 17 = (2) \cdot (5) \cdot (2 \cdot 5) \cdot (17) \cdot (2 \cdot 13) = 2 \cdot 5 \cdot 10 \cdot 17 \cdot 26$

$so n = 5$

Solution

Q.
Let $z = (1 + i) (1 + 2 i) (1 + 3 i)$ ... $(1 + n i),$ where $i = \sqrt{- 1}$ . If $| z |^{2} = 44200$ , then $n$ is equal to _________. [2026]

Ans.
(5)

$| z |^{2}$ $= 2^{3} \cdot 5^{2} \cdot 13 \cdot 17$

$\prod_{r = 1}^{n} (1 + r^{2}) = 2^{3} \cdot 5^{2} \cdot 13 \cdot 17 = (2) \cdot (5) \cdot (2 \cdot 5) \cdot (17) \cdot (2 \cdot 13) = 2 \cdot 5 \cdot 10 \cdot 17 \cdot 26$

$so n = 5$

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Solution

Q. Let z=(1+i)(1+2i)(1+3i)...(1+ni), where i=-1. If |z|2=44200, then n is equal to _________. [2026]

Ans. (5) |z|2=23·52·13·17 ∏r=1n(1+r2)=23·52·13·17=(2)·(5)·(2·5)·(17)·(2·13)=2·5·10·17·26 so n=5

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Q.
Let $z = (1 + i) (1 + 2 i) (1 + 3 i)$ ... $(1 + n i),$ where $i = \sqrt{- 1}$ . If $| z |^{2} = 44200$ , then $n$ is equal to _________. [2026]

Ans.
(5)

$| z |^{2}$ $= 2^{3} \cdot 5^{2} \cdot 13 \cdot 17$

$\prod_{r = 1}^{n} (1 + r^{2}) = 2^{3} \cdot 5^{2} \cdot 13 \cdot 17 = (2) \cdot (5) \cdot (2 \cdot 5) \cdot (17) \cdot (2 \cdot 13) = 2 \cdot 5 \cdot 10 \cdot 17 \cdot 26$

$so n = 5$