Advertisements
Advertisements
प्रश्न
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
Advertisements
उत्तर
We know that, \[i + i^2 + i^3 + i^4 = i - 1 - i + 1 = 0\]
\[\therefore i + i^2 + i^3 + . . . . + i^{1000} \]
\[ = \left( i + i^2 + i^3 + i^4 \right) + \left( i^5 + i^6 + i^7 + i^8 \right) + . . . + \left( i^{997} + i^{998} + i^{999} + i^{1000} \right)\]
\[ = \left( i + i^2 + i^3 + i^4 \right) + \left( i^4 i + i^4 i^2 + i^4 i^3 + i^4 i^4 \right) + . . . + \left[ \left( i^4 \right)^{249} i + \left( i^4 \right)^{249} i^2 + \left( i^4 \right)^{249} i^3 + \left( i^4 \right)^{249} i^4 \right]\]
\[ = \left( i + i^2 + i^3 + i^4 \right) + \left( i + i^2 + i^3 + i^4 \right) + . . . + \left( i + i^2 + i^3 + i^4 \right)\]
\[ = 0\]
Thus, the sum of the series
\[i + i^2 + i^3 + . . . .\] upto 1000 terms is 0.
APPEARS IN
संबंधित प्रश्न
Evaluate the following:
i457
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Show that 1 + i10 + i20 + i30 is a real number.
Express the following complex number in the standard form a + i b:
\[(1 + i)(1 + 2i)\]
Express the following complex number in the standard form a + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
Find the real value of x and y, if `((1+i)x-2i)/(3+i) + ((2-3i)y+i)/(3-i) = i, xy ∈ R, i = sqrt-1`
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
Solve the equation \[\left| z \right| = z + 1 + 2i\].
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
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 in polar form.
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
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:
`(- sqrt(5) + 2sqrt(-4)) + (1 -sqrt(-9)) + (2 + 3"i")(2 - 3"i")`
Show that `(-1 + sqrt(3)"i")^3` is a real number
Evaluate the following : i888
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
State True or False for the following:
2 is not a complex number.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
