SATHEE: Complex Numbers 1 Question 8

Complex Numbers 1 Question 8

8.

The value of $sum \sum_{n = 1}^{13} (i^{n} + i^{n + 1})$ , where $i = \sqrt{- 1}$ , equals

(a) $i$

(b) $i - 1$

(d) 0

(1998, 2M)

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Answer:

Correct Answer: 8. (b)

Solution:

$\sum_{n = 1}^{13} (i^{n} + i^{n + 1}) = \sum_{n = 1}^{13} i^{n} (1 + i) = (1 + i) \sum_{n = 1}^{13} i^{n}$

$\begin{aligned} = (1 + i) (i + i^{2} + i^{3} + \dots + i^{13}) = (1 + i) (\frac{i - (1 - i^{13})}{1 - i}) \\ = (1 + i) (\frac{i (1 - i)}{1 - i}) = (1 + i) i = i - 1 \end{aligned}$

Alternate Solution

Since, sum of any four consecutive powers of iota is zero.

$∴ \sum_{n = 1}^{13} (i^{n} + i^{n + 1})$

$= (i + i^{2} + \dots + i^{13}) + (i^{2} + i^{3} + \dots + i^{14})$

$= i + i^{2} = i - 1$

Complex Numbers 1 Question 8

8.

Answer:

Alternate Solution

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Complex Numbers 1 Question 8

8.

Answer:

Alternate Solution

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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