For any integer k , let k = cos ( k 7 ) + i sin ( k 7 ) , where i = - 1 . The value of the expression k = 1 12 | a k + 1 - a k | k = 1 3 | a 4 k - 1 - a 4 k - 2 | is [2015]

Question

For any integer $k$ , let $α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}),$ where $i = \sqrt{- 1}$ . The value of the expression $\frac{\sum_{k = 1}^{12} || α_{k + 1} - α_{k} ||}{\sum_{k = 1}^{3} || α_{4 k - 1} - α_{4 k - 2} ||}$ is [2015]

Smart Achievers · Accepted Answer

(4)

$Given: α_{k} = \cos \frac{k π}{7} + i \sin \frac{k π}{7} = e^{\frac{i π k}{7}}$

$α_{k + 1} - α_{k} = e^{\frac{i π (k + 1)}{7}} - e^{\frac{i π k}{7}} = e^{\frac{i π k}{7}} (e^{\frac{i π}{7}} - 1)$

$| α_{k + 1} - α_{k} |$ $=$ $| e^{\frac{i π}{7}} - 1 |$

$\Rightarrow \sum_{k = 1}^{12}$ $| α_{k + 1} - α_{k} |$ $= 12$ $| e^{\frac{i π}{7}} - 1 |$

$Similarly, \sum_{k = 1}^{3}$ $| α_{4 k - 1} - α_{4 k - 2} |$ $= 3$ $| e^{\frac{i π}{7}} - 1 |$

$∴$ $\frac{\sum_{k = 1}^{12} || α_{k + 1} - α_{k} ||}{\sum_{k = 1}^{3} || α_{4 k - 1} - α_{4 k - 2} ||} = 4$

Solution

Q.
For any integer $k$ , let $α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}),$ where $i = \sqrt{- 1}$ . The value of the expression $\frac{\sum_{k = 1}^{12} || α_{k + 1} - α_{k} ||}{\sum_{k = 1}^{3} || α_{4 k - 1} - α_{4 k - 2} ||}$ is [2015]

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Solution

Q. For any integer k, let αk=cos(kπ7)+isin(kπ7), where i=-1. The value of the expression ∑k=112|αk+1-αk|∑k=13|α4k-1-α4k-2| is [2015]

Ans. (4) Given: αk=coskπ7+isinkπ7=eiπk7 αk+1-αk=eiπ(k+1)7-eiπk7=eiπk7(eiπ7-1) |αk+1-αk|=|eiπ7-1| ⇒∑k=112|αk+1-αk|=12|eiπ7-1| Similarly, ∑k=13|α4k-1-α4k-2|=3|eiπ7-1| ∴∑k=112|αk+1-αk|∑k=13|α4k-1-α4k-2|=4

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Q.
For any integer $k$ , let $α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}),$ where $i = \sqrt{- 1}$ . The value of the expression $\frac{\sum_{k = 1}^{12} || α_{k + 1} - α_{k} ||}{\sum_{k = 1}^{3} || α_{4 k - 1} - α_{4 k - 2} ||}$ is [2015]