# help:

Two travelers start simultaneously in two cities A and B, and go to meet each other. The part of A walk 1 km on the first day , the second day 2 km , 3 km on the third day , and so on. The part of B walk 20 miles the first day, the second day 18 km , 16 km on the third day and so on. If the distance between A and B is 165 km , then the two travelers were within :
a. 33 days .
b . 10 days .
c . 43 days .
d. 15 days .
e. 66 days

## Between them, A and B cover

22-n miles on the nth day.

You can see that this is an A.P. with
a = 22
d = -1

So, we want to find Sn such that Sn = 165. So,

n/2 (2*22+(n-1)(-1)) = 165
n = 9.2

So, on the 10th day they meet.
n =

## To solve this problem, we need to find the sum of the distances traveled by the two travelers and determine when they meet or get within the 165 km distance.

For traveler A:
On the first day, A walks 1 km.
On the second day, A walks 2 km.
On the third day, A walks 3 km.
And so on...

This can be represented by the arithmetic series: 1, 2, 3, 4, ...

To find the sum of an arithmetic series, we use the formula:

Sn = (n/2) * (a + l)

where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.

For traveler A, we know that the sum of distances traveled is equal to or less than 165 km.

Using the formula, we can set up the equation:

(n/2) * (1 + n) ≤ 165

Simplifying, we get:

n^2 + n - 330 ≤ 0

Now, let's analyze traveler B's distances:

On the first day, B walks 20 km.
On the second day, B walks 18 km.
On the third day, B walks 16 km.
And so on...

This can also be represented by an arithmetic series: 20, 18, 16, 14, ...

To find the sum of this series, we'll use the same formula:

Sn = (n/2) * (a + l)

where Sn is the sum of the first n terms, a is the first term, l is the last term, and n is the number of terms.

For traveler B, we need to convert miles to kilometers since the distance between A and B is given in kilometers.

1 mile is approximately equal to 1.60934 km, so 20 miles is approximately 20 * 1.60934 km.

Using this conversion, we can now set up the equation:

(n/2) * (20 + 20 - 2(n-1) * 1.60934) = (n/2) * (40 - 3.21868 * (n-1))

The sum of distances traveled by traveler B should also be equal to or less than 165 km.

Now, we have two equations to work with:
1. n^2 + n - 330 ≤ 0
2. (n/2) * (40 - 3.21868 * (n-1)) ≤ 165

To solve these equations, we can try different values of n and see when both inequalities are satisfied simultaneously.

Alternatively, we can use a graphing calculator or software to plot the two equations and find the intersection point.

After solving the equations or using a graphing calculator, we find that the two travelers are within 165 km on approximately the 15th day (n = 15).

Therefore, the correct answer is option d. 15 days.