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Charlie and his friends are playing a game. They started round 1 of the game with 27 people. The number of people in the first five rounds is shown in the table.

Round Number of People
1 27
2 25
3 23
4 21
5 19
What is the explicit function that can be used to determine the number of people remaining in the
round?



Answer :

ynahayuri
To determine the explicit function that can be used to determine the number of people remaining in each round, we need to observe the pattern in the given data. Let's look at the number of people in each round:

- Round 1: 27 people
- Round 2: 25 people
- Round 3: 23 people
- Round 4: 21 people
- Round 5: 19 people

We can see that the number of people decreases by 2 in each successive round. This suggests a linear relationship.

A linear function can be written in the form:
\[ P(n) = P_1 - d(n - 1) \]
where:
- \( P(n) \) is the number of people in round \( n \),
- \( P_1 \) is the number of people in the first round,
- \( d \) is the common difference (the amount by which the number of people decreases each round),
- \( n \) is the round number.

From the data, we have:
- \( P_1 = 27 \) (the number of people in the first round),
- \( d = 2 \) (the decrease in the number of people per round).

Substituting these values into the linear function formula:
\[ P(n) = 27 - 2(n - 1) \]

Simplifying the expression:
\[ P(n) = 27 - 2n + 2 \]
\[ P(n) = 29 - 2n \]

Therefore, the explicit function that can be used to determine the number of people remaining in any round \( n \) is:
\[ P(n) = 29 - 2n \]

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