A and B set out to meet each other from two places 165 km apart. A travels 15 km the first day, 14 km the second day, 13 km the third day and so on. B travels 10 km the first day, 12 km the second day, 14 km the third day and so on. After how many days will they meet?

Options

A. 8 days
B. 5 days
C. 6 days
D. 7 days

Answer

To solve the problem step by step, we need to analyze the distances traveled by A and B each day and find out after how many days they will meet.

### Step 1: Understand the distance traveled by A and B
- A travels:
- Day 1: 15 km
- Day 2: 14 km
- Day 3: 13 km
- ...
- This forms a sequence where A's distance decreases by 1 km each day.

- B travels:
- Day 1: 10 km
- Day 2: 12 km
- Day 3: 14 km
- ...
- This forms a sequence where B's distance increases by 2 km each day.

### Step 2: Write down the distances for A and B
- The distance A travels on the nth day can be expressed as:
\[ A_n = 15 - (n - 1) = 16 - n \]
- The distance B travels on the nth day can be expressed as:
\[ B_n = 10 + 2(n - 1) = 8 + 2n \]

### Step 3: Calculate the total distance covered by A and B each day
- The total distance covered by A and B on the nth day is:
\[ D_n = A_n + B_n = (16 - n) + (8 + 2n) = 24 + n \]

### Step 4: Find the sum of distances until they meet
- They will meet when the total distance covered equals the distance between them, which is 165 km. We need to find the sum of the distances covered over n days:
\[ S_n = D_1 + D_2 + D_3 + ... + D_n \]
- The distances for the first few days are:
- Day 1: \(D_1 = 24 + 1 = 25\)
- Day 2: \(D_2 = 24 + 2 = 26\)
- Day 3: \(D_3 = 24 + 3 = 27\)
- ...
- This forms an arithmetic progression (AP) with the first term \(a = 25\) and common difference \(d = 1\).

### Step 5: Use the formula for the sum of an arithmetic series
- The sum of the first n terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
- Plugging in the values:
\[ S_n = \frac{n}{2} \times (2 \times 25 + (n - 1) \times 1) = \frac{n}{2} \times (50 + n - 1) = \frac{n}{2} \times (n + 49) \]

### Step 6: Set the sum equal to 165 km
- We set the equation:
\[ \frac{n}{2} \times (n + 49) = 165 \]
- Multiplying both sides by 2 gives:
\[ n(n + 49) = 330 \]
- Rearranging gives:
\[ n^2 + 49n - 330 = 0 \]

### Step 7: Solve the quadratic equation
- We can factor the quadratic:
\[ (n - 6)(n + 55) = 0 \]
- This gives us two solutions:
\[ n = 6 \quad \text{or} \quad n = -55 \]
- Since n cannot be negative, we have:
\[ n = 6 \]

### Conclusion
A and B will meet after **6 days**.