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
Express the given complex number in the form a + ib: i9 + i19
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Find the value of the following expression:
i + i2 + i3 + i4
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:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan 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 the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
Write the argument of −i.
Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
Simplify : `sqrt(-16) + 3sqrt(-25) + sqrt(-36) - sqrt(-625)`
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
(1 + 2i)(– 2 + i)
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
(1 + i)(1 − i)−1
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
(1 + i)−3
Evaluate the following : i35
Evaluate the following : i116
Evaluate the following : i–888
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
State True or False for the following:
2 is not a complex number.
