Inner Summation Evaluation
The inner summation is $S_{inner} = \sum_{r=1}^{10} \left( \sin \frac{2r\pi}{11} - i \cos \frac{2r\pi}{11} \right)$.
We can rewrite the term inside the summation:
$\sin \frac{2r\pi}{11} - i \cos \frac{2r\pi}{11} = -i \left( \cos \frac{2r\pi}{11} + i \sin \frac{2r\pi}{11} \right) = -i e^{i \frac{2r\pi}{11}}$
Let $\omega = e^{i \frac{2\pi}{11}}$. Then the expression becomes:
$S_{inner} = \sum_{r=1}^{10} -i \omega^r = -i \sum_{r=1}^{10} \omega^r$
Since $\omega$ is an 11th root of unity (i.e., $\omega^{11} = 1$), the sum of all roots is zero: $1 + \omega + \omega^2 + \dots + \omega^{10} = 0$.
Therefore, the sum from $r=1$ to $10$ is $\sum_{r=1}^{10} \omega^r = (\sum_{r=0}^{10} \omega^r) - \omega^0 = 0 - 1 = -1$.
Substituting this back, we get $S_{inner} = -i(-1) = i$.
Outer Summation Calculation
The original expression simplifies to $\sum_{K=1}^{32} (3K+2) (S_{inner})^K = \sum_{K=1}^{32} (3K+2) i^K$.
This is an arithmetic-geometric series. We can use the standard formula for $\sum_{K=1}^{n} (aK+b)x^K$:
$ \sum_{K=1}^{n} (aK+b)x^K = \frac{ax(1-(n+1)x^n+nx^{n+1})}{(1-x)^2} + \frac{bx(1-x^n)}{1-x} $
In this case, $a=3$, $b=2$, $n=32$, and $x=i$. First, calculate the necessary powers of $i$:
- $i^{32} = (i^4)^8 = 1^8 = 1$
- $i^{33} = i^{32} \cdot i = 1 \cdot i = i$
Next, calculate the terms needed for the formula:
- $1-x = 1-i$
- $(1-x)^2 = (1-i)^2 = 1 - 2i + i^2 = 1 - 2i - 1 = -2i$
- $1-x^n = 1 - i^{32} = 1 - 1 = 0$
- $1-(n+1)x^n+nx^{n+1} = 1 - (32+1)i^{32} + 32i^{33} = 1 - 33(1) + 32(i) = 1 - 33 + 32i = -32 + 32i = 32(i-1)$
Now, substitute these values into the formula:
$ S_{outer} = \frac{3 \cdot i \cdot (32(i-1))}{(-2i)} + \frac{2 \cdot i \cdot (0)}{(1-i)} $
$ S_{outer} = \frac{96i(i-1)}{-2i} + 0 $
Simplify the expression:
$ S_{outer} = -48(i-1) $
$ S_{outer} = -48i + 48 = 48(1-i) $
Final Expression Result
The value of the given expression is $48(1-i)$.
Similar Questions
- If $\alpha$, $\beta$ are the roots of the equation $x^2 - px + q = 0$ and $\alpha > 0$, $\beta > 0$, then $\alpha^{\frac{1}{4}} + \beta^{\frac{1}{4}} = \left(p + 6\sqrt{p} + 4q^{\frac{1}{4}}\sqrt{p+2\sqrt{q}}\right)^K$, where $K$ is
- If $t_n$ denotes the $n$th term of an A.P. and $t_p = \frac{1}{q}, t_q = \frac{1}{p}$, then which one of the following options is a root of the equation $(p+2q-3r)x^2 + (q+2r-3p)x + (r+2p-3q) = 0$?
- If $f(x) = \frac{1+x}{1-x}$ and $A$ is a matrix such that $A^3 = 0$, then $f(A) =$
- If $0 < \alpha < \beta < \gamma < \frac{\pi}{2}$, then the equation $\frac{1}{x - \sin \alpha} + \frac{1}{x - \sin \beta} + \frac{1}{x - \sin \gamma} = 0$ has
- On the set $\mathbb{R}$ of real numbers the relation $\rho$, defined by $x \rho y$ $(x, y \in \mathbb{R})$ iff
- Let $a_1, a_2, a_3, ...$ are in G.P. such that $n > m, a_n > a_m$ and $a_1 + a_n = 66, a_2 \cdot a_{n-1} = 128$. If $\sum_{r=1}^n a_r = 126$, then $n$ is
- Let 10 Bags $B_1, B_2, ..., B_{10}$ which contains $21, 22, ..., 30$ different articles respectively. Then the total number of ways to bring out 10 articles from a Bag is
- The total number of polynomials of the form $x^3 + ax^2 + bx + c$ which is divisible by $x^2 + 1$, where $a, b, c \in \{1, 2, 3, ..., 10\}$ is
- The term independent of $x$ in the expansion of $\left(\frac{x+1}{x^{\frac{2}{3}} - x^{\frac{1}{3}} + 1} - \frac{x-1}{x - x^{\frac{1}{2}}}\right)^{15}$ is equal to
- The equation $|x+1|^{\log_{x+1}(3+2x-x^2)} = (x-3)|x|$ has
Important Questions from Algebra
- Let A be a $3 \times 3$ matrix such that $A + A^T = O$. If $A\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}$, $A^2\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix}$ and $\det(adj(2 \ adj(A + I))) = (2)^\alpha \cdot (3)^\beta \cdot (11)^\gamma$, $\alpha, \beta, \gamma$ are non-negative integers, then $\alpha + \beta + \gamma$ is equal to _________
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
- Let ABC be a triangle. Consider four points $p_1, p_2, p_3, p_4$ on the side AB, five points $p_5, p_6, p_7, p_8, p_9$ on the side BC, and four points $p_{10}, p_{11}, p_{12}, p_{13}$ on the side AC. None of these points is a vertex of the triangle ABC. Then the total number of pentagons, that can be formed by taking all the vertices from the points $p_1, p_2, ..., p_{13}$, is _________
- Let n be the number obtained on rolling a fair die. If the probability that the system
$x - ny + z = 6$
$x + (n - 2)y + (n + 1)z = 8$
$(n - 1)y + z = 1$
has a unique solution is $\frac{k}{6}$, then the sum of k and all possible values of n is : - If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is a solution of the system of equations $AX = B$, where $\text{adj } A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$, then $|x + y + z|$ is equal to :
