4 Calculate: $$ \frac{4+6+8 \ldots+806}{5+10+15+\ldots+625} $$

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UpStudy AI · Accepted Answer

To calculate the ratio $$\frac{4 + 6 + 8 + \ldots + 806}{5 + 10 + 15 + \ldots + 625}$$, we'll evaluate both the numerator and the denominator separately and then simplify the resulting fraction.

### **Numerator: $$4 + 6 + 8 + \ldots + 806$$**

This is an arithmetic sequence where:
- **First term ($$a_1$$)** = 4
- **Common difference ($$d$$)** = 2
- **Last term ($$a_n$$)** = 806

**Number of terms ($$n$$)**:
$$
a_n = a_1 + (n-1)d \
806 = 4 + (n-1) \times 2 \
806 = 4 + 2n - 2 \
806 = 2n + 2 \
804 = 2n \
n = 402
$$

**Sum of the numerator ($$S_n$$)**:
$$
S_n = \frac{n}{2} \times (a_1 + a_n) \
S_n = \frac{402}{2} \times (4 + 806) \
S_n = 201 \times 810 \
S_n = 162{,}810
$$

### **Denominator: $$5 + 10 + 15 + \ldots + 625$$**

This is also an arithmetic sequence where:
- **First term ($$a_1$$)** = 5
- **Common difference ($$d$$)** = 5
- **Last term ($$a_m$$)** = 625

**Number of terms ($$m$$)**:
$$
a_m = a_1 + (m-1)d \
625 = 5 + (m-1) \times 5 \
625 = 5m \
m = 125
$$

**Sum of the denominator ($$S_m$$)**:
$$
S_m = \frac{m}{2} \times (a_1 + a_m) \
S_m = \frac{125}{2} \times (5 + 625) \
S_m = \frac{125}{2} \times 630 \
S_m = 125 \times 315 \
S_m = 39{,}375
$$

### **Ratio of the Sums**

$$
\frac{S_n}{S_m} = \frac{162{,}810}{39{,}375}
$$

**Simplifying the Fraction:**
$$
\frac{162{,}810 \div 15}{39{,}375 \div 15} = \frac{10{,}854}{2{,}625} \
\frac{10{,}854 \div 3}{2{,}625 \div 3} = \frac{3{,}618}{875}
$$

The fraction $$\frac{3{,}618}{875}$$ cannot be simplified further as 3,618 and 875 share no common factors besides 1.

### **Final Answer**

$$
\frac{3,\!618}{875}
$$

Which is the simplified ratio of the two sums.

**Decimal Representation (Optional):**
$$
\frac{3{,}618}{875} \approx 4.1343
$$

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**Answer:**

The ratio is $$\dfrac{3,\!618}{875}$$.
The ratio is $$\frac{3,\!618}{875}$$.

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