Difference between revisions of "2023 SSMO Relay Round 1 Problems/Problem 2"
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<cmath>\begin{align*} | <cmath>\begin{align*} | ||
S = \sum_{i=0}^\infty a_i &= a_0+a_1+a_2+\sum_{i=3}^\infty a_i\\ | S = \sum_{i=0}^\infty a_i &= a_0+a_1+a_2+\sum_{i=3}^\infty a_i\\ | ||
| − | &= | + | &= 3+1+2022+\sum_{i=3}^\infty \left(a_{i-1}-\frac{a_{i-3}}{8}\right)\\ |
| − | &= | + | &= 2026+\sum_{i=3}^{\infty}a_{i-1}-\sum_{i=3}^\infty \frac{a_{i-3}}{8}\\ |
| − | &= | + | &= 2026+\sum_{i=2}^\infty a_i-\frac{\sum_{i=0}a_i}{8}\\ |
| − | &= | + | &= 2026+\left(\left[ \sum_{i=0}^\infty a_i \right]-a_0-a_1\right)-\frac{S}{8}\\ |
| − | &= | + | &= 2026+(S-3-1)-\frac{S}{8}\\ |
&= 2022+\frac{7S}{8}.\\ | &= 2022+\frac{7S}{8}.\\ | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
Latest revision as of 17:15, 15 September 2025
Problem
Let 
. Let 
, and let 
for 
Find ![\[\sum_{i=0}^\infty a_i.\]](http://latex.artofproblemsolving.com/4/9/d/49d916ca4e9ac73751ff00a7a75776bd2eaca943.png)
Solution
We have 
. Let
![\begin{align*} S = \sum_{i=0}^\infty a_i &= a_0+a_1+a_2+\sum_{i=3}^\infty a_i\\ &= 3+1+2022+\sum_{i=3}^\infty \left(a_{i-1}-\frac{a_{i-3}}{8}\right)\\ &= 2026+\sum_{i=3}^{\infty}a_{i-1}-\sum_{i=3}^\infty \frac{a_{i-3}}{8}\\ &= 2026+\sum_{i=2}^\infty a_i-\frac{\sum_{i=0}a_i}{8}\\ &= 2026+\left(\left[ \sum_{i=0}^\infty a_i \right]-a_0-a_1\right)-\frac{S}{8}\\ &= 2026+(S-3-1)-\frac{S}{8}\\ &= 2022+\frac{7S}{8}.\\ \end{align*}](http://latex.artofproblemsolving.com/8/3/7/837ae279f1014681bc91459e2ee047b926aff7f7.png)
We have
~pinkpig
