If has exactly 7 solutions in the interval

MATHEMATICS · Single correct · A Level · Mathematics · Trigonometry · Trigonometry

Problem

2\tan^2 \theta - 5\sec \theta = 1

has exactly 7 solutions in the interval

\left[0, \frac{n\pi}{2}\right]

, for the least value of

n \in \mathbb{N}

then

\sum_{k=1}^{n} \frac{k}{2^k}

is equal to: (A)

\frac{1}{2^{15}}(2^{14} - 14)

(B)

\frac{1}{2^{14}}(2^{15} - 15)

(C)

1 - \frac{15}{2^{13}}

(D)

\frac{1}{2^{13}}(2^{14} - 15)