Question

If $D_k=\left|\begin{array}{ccc}{2}^{k-1}& \frac{1}{k(k+1)}& sink\theta \\ x& y& z\\ 2^n-1& \frac{n}{n+1}& \frac{sin\left(\frac{n+1}{2}\right)\theta sin\frac{n}{2}\theta }{sin\theta /2}\end{array}\right|$ then $\sum _{k=1}^n{D}_k$ is equal to

• A. 0
• B. independent of n
• C. independent of $\theta$
• D. independent of x, y and z

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A 0
B independent of n
C independent of $\theta$
D independent of x, y and z

The (1,1) term,$=\Sigma 2^{k-1}$
$=2^0+2^1+2^2+..{.2}^{n-1}$ (Geometric series)
$=(2^n-1)$
The (1,2) term,$=\Sigma \frac{1}{k(k+1)}$$=\Sigma (\frac{1}{k}-\frac{1}{k+1})$
$=1-\frac{1}{n+1}$
$=\frac{n}{n+1}$
The (1,3) term,
$=\Sigma sink\theta =\frac{sin\left(\frac{n+1}{2}\right)\theta sin\frac{n}{2}\theta }{sin\frac{\theta }{2}}$
As $R_1=R_3$ so, $\Delta =0$ and is independent of any parameters.