Trigonometry
Telescoping Sums
MMTS_Full_Test_03
Grade 12
Question:
If $\sum_{r=1}^n\left(\dfrac{\tan 2^{r-1}}{\cos 2^r}\right)=\tan p^n-\tan q$, then find the value of $(p+q)$
Step-by-Step Solution
Key Concept: Use identity $\tan\theta/\cos 2\theta=(\tan 2\theta-\tan\theta)/2$
The sum telescopes to $\tan(2^n\cdot 1)-\tan(1)$... So $p=2, q=1$... but wait. Actually the sum $=\frac{1}{2}(\tan 2^n - \tan 1)$... $p+q=3$.
Correct Answer: 3