Question

(1+I) (1+2i) (1+3i),,,,,,,(1+ni)=x2+y2, show that 2,5,10,,,,,(1+n2)=x2+y2

Answer 1

Correct Question:

If (1 + i) (1 + 2i) (1 + 3i) ... (1 + ni) = x + yi, show that (2) (5) (10) ... (1 + n²) = x² + y²

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Answer:

Given:

• (1 + i) (1 + 2i) (1 + 3i) ... (1 + ni) = x + yi

Solution:

(1 + i) (1 + 2i) (1 + 3i) ... (1 + ni) = x + yi ... (1)

It's conjugate pair can be given by

(1 - i) (1 - 2i) (1 - 3i) ... (1 - ni) = x - yi ... (2)

Multiply equation (1) and (2), we get

(1² - i²) (1² - 2²i²) (1² - 3²i²) ... (1 - n²i²) = x² - y²i²

[By using identity, (a + b) (a - b) = a² - b²]

Substitute i² = - 1, we get

=> (2) (5) (10) ... (1 + n²) = x² + y²

Hence, proved.