Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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Find the Value of the Following Expression:1+ I2 + I4 + I6 + I8 + ... + I20 - Mathematics

Find the value of the following expression:

1+ i2 + i4 + i6 + i8 + ... + i20

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Solution

\[(vi) 1 + i^2 + i^4 + i^6 + i^8 + . . . + i^{20} \]

\[ \because i^2 = - 1, \]

\[ i^4 = 1, \]

\[ i^6 = - 1, \]

\[ i^8 = 1, \]

\[ i^{20} = 1\]

\[ \therefore 1 + i^2 + i^4 + i^6 + i^8 + . . . + i^{20} \]

\[ = \left[ 1 + \left( - 1 \right) \right] + \left[ 1 + \left( - 1 \right) \right] + \left[ 1 + \left( - 1 \right) \right] + . . . + \left[ 1 + \left( - 1 \right) \right] + 1\]

\[ = 5 \times \left[ 1 + \left( - 1 \right) \right] + 1 \left[ \text { As, there are 11 terms} \right]\]

\[ = 5 \times 0 + 1\]

\[ = 1\]

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Exercise 13.1 | Q 3.6 | Page 4
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