Advertisements
Advertisements
Question
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Advertisements
Solution
\[\frac{i^{4n + 1} - i^{4n - 1}}{2}\]
\[ = \frac{i - \frac{1}{i}}{2} \left( \because i^{4n} = 1, i^{- 1} = \frac{1}{i} \right)\]
\[ = \frac{\frac{i^2 - 1}{i}}{2}\]
\[ = \frac{i^2 - 1}{2i}\]
\[ = \frac{- 1 - 1}{2i}\]
\[ = \frac{- 2}{- 2i} \]
\[ = \frac{- 1}{i}\]
\[ = \frac{- i}{i^2} \left( \because i^2 = - 1 \right)\]
\[ = \frac{- i}{- 1}\]
\[ = i\]
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Show that 1 + i10 + i20 + i30 is a real number.
Express the following complex number in the standard form a + i b:
\[\frac{1}{(2 + i )^2}\]
Express the following complex number in the standard form a + i b:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
Find the multiplicative inverse of the following complex number:
1 − i
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Im `(1/(z_1overlinez_1))`
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
If \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
Express the following complex in the form r(cos θ + i sin θ):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
If z1 and z2 are two complex numbers such that \[\left| z_1 \right| = \left| z_2 \right|\] and arg(z1) + arg(z2) = \[\pi\] then show that \[z_1 = - \bar{{z_2}}\].
Write −1 + i \[\sqrt{3}\] in polar form .
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
The polar form of (i25)3 is
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
If z is a complex number, then
Find a and b if (a – b) + (a + b)i = a + 5i
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`(3 + 2"i")/(2 - 5"i") + (3 -2"i")/(2 + 5"i")`
Evaluate the following : i888
Evaluate the following : i93
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
State True or False for the following:
The order relation is defined on the set of complex numbers.
Show that `(-1 + sqrt3 "i")^3` is a real number.
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
Show that `(-1+ sqrt(3)i)^3` is a real number.
