Complex Numbers 2 Question 16
16. For positive integers $n _1, n _2$ the value of expression $(1+i)^{n _1}+\left(1+i^{3}\right)^{n _1}+\left(1+i^{5}\right)^{n _2}+\left(1+i^{7}\right)^{n _2}$, here $i=\sqrt{-1}$ is a real number, if and only if
$(1996,2 M)$
(a) $n _1=n _2+1$
(b) $n _1=n _2-1$
(c) $n _1=n _2$
(d) $n _1>0, n _2>0$
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Answer:
Correct Answer: 16. (b)
Solution:
- $(1+i)^{n _1}+(1-i)^{n _1}+(1+i)^{n _2}+(1-i)^{n _2}$
$$ \begin{aligned} & =\left[{ }^{n _1} C _0+{ }^{n _1} C _1 i+{ }^{n _1} C _2 i^{2}+{ }^{n _1} C _3 i^{3}+\ldots\right] \\ & +\left[{ }^{n _1} C _0-{ }^{n _1} C _1 i+{ }^{n _1} C _2 i^{2}-{ }^{n _1} C _3 i^{3}+\ldots\right] \\ & +\left[{ }^{n _2} C _0+{ }^{n _2} C _1 i+{ }^{n _2} C _2 i^{2}+{ }^{n _2} C _3 i^{3}+\ldots\right] \\ & +\left[{ }^{n _2} C _0-{ }^{n _2} C _1 i+{ }^{n _2} C _2 i^{2}-{ }^{n _2} C _3 i^{3}+. .\right] \\ & =2\left[{ }^{n _1} C _0+{ }^{n _1} C _2 i^{2}+{ }^{n _1} C _4 i^{4}+\ldots\right] \\ & +2\left[{ }^{n _2} C _0+{ }^{n _2} C _2 i^{2}+{ }^{n _2} C _4 i^{4}+\ldots\right] \\ & =2\left[{ }^{n _1} C _0-{ }^{n _1} C _2+{ }^{n _1} C _4-\ldots\right]+2\left[{ }^{n _2} C _0-{ }^{n _2} C _2\right. \\ & \left.+{ }^{n _2} C _4-\ldots\right] \end{aligned} $$
This is a real number irrespective of the values of $n _1$ and $n _2$.
Alternate Solution
${(1+i)^{n _1}+(1-i)^{n _1} }+{(1+i)^{n _2}+(1-i)^{n _2} }$
$\Rightarrow$ A real number for all $n _1$ and $n _2 \in R$.
$\left[\because z+\bar{z}=2 \operatorname{Re}(z) \Rightarrow(1+i)^{n _1}+(1-i)^{n _1}\right.$ is real number for all $n \in R]$
