When expressed as a common fraction, what is the value of (2 + 4 + 6 + 8 + ... + 1340 + 1342)/(3 + 6 + 9 + 12 + ... + 2010 + 2013)?

Padewolofoofy Dec 20, 2023

#1**+1 **

First, we recognize that both the numerator and denominator are arithmetic series.

The numerator is an arithmetic series with first term a1=2, common difference d=2, and last term an =1342.

The denominator is an arithmetic series with first term a1=3, common difference d=3, and last term an=2013.

To find the number of terms in each series, we can use the formula for the $n$th term of an arithmetic series: an=a1+(n−1)d.

For the numerator: 1342=2+(n−1)(2)⇒n=671.

For the denominator: 2013=3+(n−1)(3)⇒n=671.

Now, we can use the formula for the sum of an arithmetic series: Sn=2n(a1+an).

The sum of the numerator is Sn=2671(2+1342)=450546.

The sum of the denominator is Sn=2671(3+2013)=673546.

Therefore, the value of the fraction is 450546/673546.

We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2. So, the final answer is 225273/336773.

BuiIderBoi Dec 20, 2023