SOLUTION: Simplify \frac{1 + 3 + 5 + ... + 1999 + 2001 + 2003}{2 + 4 + 6 + ... + 2000 + 2002 + 2004 + 2006 + 2008 + 2010}.

Question 1209812: Simplify \frac{1 + 3 + 5 + ... + 1999 + 2001 + 2003}{2 + 4 + 6 + ... + 2000 + 2002 + 2004 + 2006 + 2008 + 2010}.
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let $N = 1 + 3 + 5 + \dots + 2003$ and $D = 2 + 4 + 6 + \dots + 2010$.
We want to find $\frac{N}{D}$.
The numerator $N$ is the sum of an arithmetic progression with first term $a_1 = 1$, last term $a_n = 2003$, and common difference $d = 2$.
To find the number of terms $n$, we use the formula $a_n = a_1 + (n-1)d$:
$2003 = 1 + (n-1)2$
$2002 = 2(n-1)$
$1001 = n-1$
$n = 1002$
The sum of an arithmetic series is given by $S_n = \frac{n(a_1 + a_n)}{2}$.
So, $N = \frac{1002(1+2003)}{2} = \frac{1002(2004)}{2} = 1002(1002) = 1004004$.
The denominator $D$ is the sum of an arithmetic progression with first term $b_1 = 2$, last term $b_m = 2010$, and common difference $d = 2$.
To find the number of terms $m$, we use the formula $b_m = b_1 + (m-1)d$:
$2010 = 2 + (m-1)2$
$2008 = 2(m-1)$
$1004 = m-1$
$m = 1005$
The sum of an arithmetic series is given by $S_m = \frac{m(b_1 + b_m)}{2}$.
So, $D = \frac{1005(2+2010)}{2} = \frac{1005(2012)}{2} = 1005(1006) = 1011030$.
Then $\frac{N}{D} = \frac{1004004}{1011030}$.
We can simplify this fraction by dividing both numerator and denominator by their greatest common divisor.
Let's divide both by 6:
$\frac{1004004}{6} = 167334$
$\frac{1011030}{6} = 168505$
So $\frac{N}{D} = \frac{167334}{168505}$.
Let's check if there are any more common factors.
We can use the Euclidean algorithm to find the gcd of 167334 and 168505.
$168505 = 1(167334) + 1171$
$167334 = 142(1171) + 832$
$1171 = 1(832) + 339$
$832 = 2(339) + 154$
$339 = 2(154) + 31$
$154 = 4(31) + 30$
$31 = 1(30) + 1$
The gcd is 1, so the fraction is already in its simplest form.
Thus, $\frac{N}{D} = \frac{1004004}{1011030} = \frac{167334}{168505}$.
Final Answer: The final answer is $\boxed{\frac{167334}{168505}}$

RELATED QUESTIONS
2. The number of touchdowns scored by the Pittsburgh Steelers during the regular season... (answered by solver91311)

2. The number of touchdowns scored by the Pittsburgh Steelers during the regular season... (answered by solver91311,MathTherapy)

2. The number of touchdowns scored by the Pittsburgh Steelers during the regular season... (answered by solver91311)

What was the average rate of change of population between 2002 and 2004, and between 2002 (answered by ewatrrr)

The following information was taken from Johnson & Johnson annual reports. The principal... (answered by jamiya)

List of Yu-Gi-Oh! episodes From Wikipedia, the free encyclopedia Yu-Gi-Oh!... (answered by ikleyn)

Hi, Is my answer correct? 24,280 The following frequency table contains the... (answered by ewatrrr)

IS MY ANSWER CORRECT? 24,280 The following frequency table contains the total... (answered by stanbon)