$$ \frac{3+6+9+...+402}{2+4+6+...+402} $$

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UpStudy AI · Accepted Answer

To solve the expression $$ \frac{3+6+9+\ldots+402}{2+4+6+\ldots+402} $$, we will first identify the series in the numerator and the denominator.

### Step 1: Analyze the Numerator
The numerator is the sum of the series $$ 3 + 6 + 9 + \ldots + 402 $$. This is an arithmetic series where:
- The first term $$ a = 3 $$
- The common difference $$ d = 3 $$
- The last term $$ l = 402 $$

To find the number of terms $$ n $$ in this series, we can use the formula for the $$ n $$-th term of an arithmetic series:
$$
l = a + (n-1)d
$$
Substituting the known values:
$$
402 = 3 + (n-1) \cdot 3
$$
$$
402 = 3 + 3n - 3
$$
$$
402 = 3n
$$
$$
n = \frac{402}{3} = 134
$$

Now, we can find the sum $$ S_n $$ of the first $$ n $$ terms of an arithmetic series using the formula:
$$
S_n = \frac{n}{2} \cdot (a + l)
$$
Substituting the values we found:
$$
S_n = \frac{134}{2} \cdot (3 + 402) = 67 \cdot 405 = 27135
$$

### Step 2: Analyze the Denominator
The denominator is the sum of the series $$ 2 + 4 + 6 + \ldots + 402 $$. This is also an arithmetic series where:
- The first term $$ a = 2 $$
- The common difference $$ d = 2 $$
- The last term $$ l = 402 $$

To find the number of terms $$ n $$ in this series, we use the same formula:
$$
l = a + (n-1)d
$$
Substituting the known values:
$$
402 = 2 + (n-1) \cdot 2
$$
$$
402 = 2 + 2n - 2
$$
$$
402 = 2n
$$
$$
n = \frac{402}{2} = 201
$$

Now, we can find the sum $$ S_n $$ of the first $$ n $$ terms of this arithmetic series:
$$
S_n = \frac{n}{2} \cdot (a + l)
$$
Substituting the values we found:
$$
S_n = \frac{201}{2} \cdot (2 + 402) = 100.5 \cdot 404 = 40602
$$

### Step 3: Calculate the Final Expression
Now we can substitute the sums we found into the original expression:
$$
\frac{3 + 6 + 9 + \ldots + 402}{2 + 4 + 6 + \ldots + 402} = \frac{27135}{40602}
$$

### Step 4: Simplify the Fraction
To simplify $$ \frac{27135}{40602} $$, we can find the greatest common divisor (GCD) of the two numbers.

Calculating the GCD:
$$
\text{GCD}(27135, 40602) = 27
$$

Now we can simplify:
$$
\frac{27135 \div 27}{40602 \div 27} = \frac{1005}{1500}
$$

Further simplifying:
$$
\frac{1005 \div 15}{1500 \div 15} = \frac{67}{100}
$$

### Final Answer
Thus, the final result is:
$$
\frac{67}{100}
$$
The final result is $$ \frac{67}{100} $$.

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