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JEE Main · 2026 · 24 Jan 2026 · Shift 2 · Complex Numbers

Let

z = (1 + i) (1 + 2 i) (1 + 3 i) \dots (1 + ni)

, where i

= - 1

. If

∣ z ∣^{2} = 44200

, then

n

is equal to ______.

Answer: 5

Solution

Since the modulus of a product is the product of the moduli,

∣ z ∣^{2} = \prod_{r = 1}^{n} ∣1 + r i ∣^{2} = \prod_{r = 1}^{n} (1 + r^{2})

. The successive factors are

2, 5, 10, 17, 26

, and

2 \cdot 5 \cdot 10 \cdot 17 \cdot 26 = 44200

. Therefore

n = 5

.

JEE Main · 2026 · 24 Jan 2026 · Shift 2 · Integer answer

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