Question:
If
\[
\sum_{k=0}^{440} i^k=x+iy,
\]
then
\[
x^{100}+x^{99}y+x^{242}y^2+x^{97}y^3=
\]
If
\[
\sum_{k=0}^{440} i^k=x+iy,
\]
then
\[
x^{100}+x^{99}y+x^{242}y^2+x^{97}y^3=
\]
Show Hint
The powers of \(i\) repeat every \(4\) terms:
\[
1,\; i,\; -1,\; -i
\]
Use this cyclic property to simplify large powers and summations involving \(i\).
Updated On: Jun 22, 2026
- \(0\)
- \(-4\)
- \(4\)
- \(1\)
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Use the cyclic property of powers of \(i\).
The powers of \(i\) repeat in cycles of \(4\): \[ i^0=1,\quad i^1=i,\quad i^2=-1,\quad i^3=-i \] and then the pattern repeats.
Also, \[ 1+i-1-i=0 \] Hence, every block of \(4\) consecutive powers sums to zero.
Step 2: Simplify the summation.
Given, \[ \sum_{k=0}^{440} i^k \] Since \[ 440=4\times110, \] the terms from \(i^0\) to \(i^{439}\) form \(110\) complete cycles whose sum is zero.
Thus only the last term remains: \[ i^{440} \] Now, \[ i^{440}=(i^4)^{110}=1^{110}=1 \] Therefore, \[ x+iy=1 \] Comparing real and imaginary parts, \[ x=1,\qquad y=0 \]
Step 3: Substitute the values of \(x\) and \(y\).
We need to evaluate \[ x^{100}+x^{99}y+x^{242}y^2+x^{97}y^3 \] Substituting \[ x=1,\qquad y=0, \] we get \[ 1^{100}+1^{99}(0)+1^{242}(0)^2+1^{97}(0)^3 \] \[ =1+0+0+0 \] \[ =1 \]
Step 4: Final conclusion.
Hence, \[ \boxed{1} \] which corresponds to option (4).
The powers of \(i\) repeat in cycles of \(4\): \[ i^0=1,\quad i^1=i,\quad i^2=-1,\quad i^3=-i \] and then the pattern repeats.
Also, \[ 1+i-1-i=0 \] Hence, every block of \(4\) consecutive powers sums to zero.
Step 2: Simplify the summation.
Given, \[ \sum_{k=0}^{440} i^k \] Since \[ 440=4\times110, \] the terms from \(i^0\) to \(i^{439}\) form \(110\) complete cycles whose sum is zero.
Thus only the last term remains: \[ i^{440} \] Now, \[ i^{440}=(i^4)^{110}=1^{110}=1 \] Therefore, \[ x+iy=1 \] Comparing real and imaginary parts, \[ x=1,\qquad y=0 \]
Step 3: Substitute the values of \(x\) and \(y\).
We need to evaluate \[ x^{100}+x^{99}y+x^{242}y^2+x^{97}y^3 \] Substituting \[ x=1,\qquad y=0, \] we get \[ 1^{100}+1^{99}(0)+1^{242}(0)^2+1^{97}(0)^3 \] \[ =1+0+0+0 \] \[ =1 \]
Step 4: Final conclusion.
Hence, \[ \boxed{1} \] which corresponds to option (4).
Was this answer helpful?
0
0
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
View More Questions
Top AP EAPCET Complex numbers Questions
- If \( z \) is a complex number such that \( |z| = 5 \) and \( \text{Re}(z) = 3 \), then the value of \( z^2 \) is:
- \( \frac{(1+i)x-2i}{3+i} + \frac{(2-3i)y+i}{3-i} = i \implies x+y = \)
- Let z and w be two distinct non-zero complex numbers if \(|z|^2w - |w|^2z = z - w\), then
- If 1, \(\omega, \omega^2\) denote the cube roots of unity, then, the value of \( (1-\omega+\omega^2)^5 + (1+\omega-\omega^2)^5 \) is
- The values of \(\theta\), for which \( \frac{3+2i\sin\theta}{1-2i\sin\theta} \) is real are
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions