plz solve according to sequence and series pattern

a man was trying to swim to a buoy placed at a distance of 200m out in the sea. it took him 1 min to swim 20m. then a wave pushed him back 10m and he rested for about 1min before swimming again. he continued in his way for the rest of their journey. how long would it take the man to swim to the buoy?

t x

0 0
1 20
2 10
3 30
4 20
5 40
6 30
7 50
8 40
9 60
10 50
11 70
12 60
13 80
14 70
15 90
16 80
17 100
18 90
19 110
20 100
21 120
22 110
23 130
.
.
.
40 200
averages 5 meters/min

sorry I took so long, had a grandson's hockey game in the middle of that.

To solve this problem using the concept of sequences and series, we need to find the total time it takes for the man to reach the buoy.

Let's break down the problem into smaller steps and determine the pattern:

Step 1: It takes the man 1 minute to swim 20m to the buoy.
Step 2: After swimming 20m, the man is pushed back 10m by a wave.
Step 3: The man then rests for about 1 minute before swimming again.

Based on the given information, we can see that the man's progress can be described by the following pattern:

20 - 10 - 20 - 10 - 20 - 10 - ...

We can see that the man is swimming 20m and being pushed back 10m repeatedly until he reaches the buoy. This pattern will continue until the sum of the distances covered is equal to or exceeds 200m.

Now, let's create a sequence to represent the distances covered during each swim:

a1 = 20m (first swim)
a2 = -10m (pushed back by the wave)
a3 = 20m (second swim)
a4 = -10m (pushed back by the wave)
a5 = 20m (third swim)
a6 = -10m (pushed back by the wave)
...

Since the value of a2, a4, a6, and so on is the same (-10m), we can see that this portion of the sequence can be described by the formula:

an = -10, for all n where n is even.

The first portion of the sequence, representing the distances covered during each swim, can be described by the formula:

an = 20, for all n where n is odd.

Now, let's find the number of terms needed to reach a sum of at least 200m. We can use the formula for the sum of an arithmetic series:

Sn = (n/2) * (a1 + an)

We want to find the smallest n for which Sn is greater than or equal to 200m.

Using the formula, we get:

200 <= (n/2) * (20 + -10)
200 <= (n/2) * 10
20 <= (n/2) * 1
20 <= n/2
40 <= n

So, the smallest integer value of n that satisfies this inequality is 40.

Now, let's calculate the total time taken to swim to the buoy:

The man rests for about 1 minute after each swim, so the total time taken to swim 40 times will be:

Total time = 40 * (1 + 1) + 1

= 40 * 2 + 1

= 80 + 1

= 81 minutes

Therefore, it will take the man approximately 81 minutes to reach the buoy.