Can You Solve This Southeast Asian Math Olympiad (SEAMO) Sample Problem?
It’s Actually Simpler Than You Might Think
Problem
Evaluate (1 + 3 + 5…+ 2013 + 2015) / (2 + 4 + 6… + 2014 + 2016)
Pause for a moment and try to solve this problem yourself! 🧠 🧠 ✏️
Solution
Got your answer? Ok! Here’s the solution! 3…2….1..
To solve this problem, first we need to break it down into 2 equations :
(1 + 3 + 5…+ 2013 + 2015) and (2 + 4 + 6… + 2014 + 2016)
To solve the problem, we need to first find the total amount of numbers in each equation. Instead of individually counting the numbers (which would take a very very long time), we can do something much smarter. First, let’s find the total number of terms in the sequence — (1 + 3 + 5…+ 2013 + 2015). This is an arithmetic series where the first term a = 1 and the common difference d = 2, to find the total number of terms (n) in this series, we can use the formula for the (n)-th term of an arithmetic sequence:
an=a+(n−1)d
If we set (an) as 2015, then we can concur that :
2015=1+(n−1)⋅2 ∴ 2015=1+2n−2, so 2015=2n−1, this means that 2016 = 2n ∴ n= 1008.
1008, it thus the total number of terms in the series (1 + 3 + 5…+ 2013 + 2015). Now to actually calculate the sum of the series, we can apply Gauss Summation:
1008 ⋅ (2015 + 1) = 1008 ⋅ 2016 = 1016064
1016064 is the total sum of the series. We can apply this to the next series — (2 + 4 + 6… + 2014 + 2016).
an= a + (n−1)d
a = 2 & d = 2
an = 2016
2016 = 2 + (n−1) ⋅ 2, therefore, 2016 = 2 + 2n − 2, so 2016 = 2n ∴ n = 1008
(1008/2) ⋅ 2018 = 504 ⋅ 2018 = 1018032
Now that we have the two sums of the series, we can solve the equation.
1016064 / 1018032 is the answer, however, it can be simplified further into 1008 / 1009, the final answer.
