|
|
| Line 5: |
Line 5: |
| | | | |
| | ==Solution== | | ==Solution== |
| − | We have <math>T = 2022</math>. Let
| |
| − | <cmath>\begin{align*}
| |
| − | S = \sum_{i=0}^\infty a_i &= a_0+a_1+a_2+\sum_{i=3}^\infty a_i\\
| |
| − | &= 0+1+2022+\sum_{i=3}^\infty \left(a_{i-1}-\frac{a_{i-3}}{8}\right)\\
| |
| − | &= 2023+\sum_{i=3}^{\infty}a_{i-1}-\sum_{i=3}^\infty\frac{a_{i-3}}{8}\\
| |
| − | &= 2023+\sum_{i=2}^\infty a_i-\frac{\sum_{i=0}a_i}{8}\\
| |
| − | &= 2023+\left(\sum_{i=0}^\infty a_i-a_0-a_1\right)-\frac{S}{8}\\
| |
| − | &= 2023+(S-0-1)-\frac{S}{8}\\
| |
| − | &= 2022+\frac{7S}{8}.\\
| |
| − | \end{align*}</cmath>
| |
| − | We have
| |
| − | <cmath>\begin{align*}
| |
| − | S &= 2022+\frac{7S}{8}\implies\\
| |
| − | \frac{S}{8} &= 2022\implies\\
| |
| − | S &= 8\cdot2022 = \boxed{16176}.
| |
| − | \end{align*}</cmath>
| |
| − |
| |
| − | ~pinkpig
| |
Latest revision as of 11:23, 15 September 2025
. Find the number of solutions to the equation
![\[\sec^{T} (Tx) - \tan^{T}(Tx) = 1\]](http://latex.artofproblemsolving.com/5/e/d/5ed50f1e674d74b015125e05c7737794c09c34a1.png)
such 
