Chairs in an auditorium are arranged in rows in such a way that the first two rows each have the same number of chairs. The third and fourth rows each have three more chairs than the first and second row; the fifth and sixth rows each have three more chairs than the third and fourth row, etc. The sequence of number of chairs for every second row forms an arithmetic sequence. The first two rows each have 27 chairs, and the last two rows each have 114 chairs.
How many rows of chairs are there?
How can this be? Because the first two rows have the same amount of chairs, and no whole number added by itself equals 27. I'm Confused.
The first 2 rows *each* have 27 chairs.
What we have here is the fact that each pair of rows of chairs has 3 more per row more than the previous pair.
114-27 = 87 = 3*29.
So, there are 30 pairs of rows. The first pair have 27 chairs each, and the 30th pair have 114 chairs each.
Sounds like 60 rows of chairs.
Let's break down the problem step by step to understand it better:
1. The first two rows each have the same number of chairs, which is 27. So, we can say the number of chairs in the first two rows is 27.
2. The third and fourth rows each have three more chairs than the first and second rows. This means the number of chairs in the third and fourth rows would be 27 + 3 = 30.
3. The fifth and sixth rows each have three more chairs than the third and fourth rows. So, the number of chairs in the fifth and sixth rows would be 30 + 3 = 33.
4. We can observe that the number of chairs in each second row forms an arithmetic sequence. Starting with the first two rows having 27 chairs, the third and fourth rows have 30 chairs, and the fifth and sixth rows have 33 chairs.
5. The last two rows each have 114 chairs. We can use this information to find the number of rows in total.
6. Let's find the common difference in the arithmetic sequence. The difference between the number of chairs in the first two rows and the number of chairs in the third and fourth rows is 30 - 27 = 3.
7. Since the difference between consecutive terms in an arithmetic sequence is constant, the difference between the number of chairs in the third and fourth rows and the number of chairs in the fifth and sixth rows would also be 3.
8. Continuing this pattern, the difference between the number of chairs in the n-th and (n+1)-th rows would be 3 for every second row.
9. To find the number of rows, we can set up an equation. Let's assume there are 'n' rows in total. The number of chairs in the last row would be 27 + (n/2 - 1) * 3 = 114.
10. Solving the equation, we can find the value of 'n'. Rearranging the equation, we get (n/2 - 1) * 3 = 114 - 27.
11. Simplifying, we have (n/2 - 1) * 3 = 87.
12. Dividing both sides by 3, we get (n/2 - 1) = 29.
13. Adding 1 to both sides, we have n/2 = 30.
14. Multiplying both sides by 2, we get n = 60.
Therefore, there are 60 rows of chairs in the auditorium.
To solve this problem, we need to use the given information and break it down step by step.
Let's start by understanding the pattern in the number of chairs for every second row. We are told that the difference between the number of chairs in each pair of consecutive rows is constant. In other words, the number of chairs increases by the same amount for each pair of rows.
Let's denote the number of chairs in the first two rows as "x". According to the given information, x = 27.
Next, we know that the number of chairs in the third and fourth rows is three more than the first two rows. This implies that the number of chairs in the third and fourth rows is x + 3.
Similarly, the number of chairs in the fifth and sixth rows will be three more than the third and fourth rows. Therefore, the number of chairs in the fifth and sixth rows is (x + 3) + 3, which simplifies to x + 6.
We can continue this pattern: for every second pair of rows, the number of chairs will be three more than the previous pair. So for the seventh and eighth rows, the number of chairs will be (x + 6) + 3, which simplifies to x + 9.
We are also given that the last two rows each have 114 chairs. Denoting the number of chairs in the last two rows as "y," we have y = 114.
Now, let's find the value of x. We know that x = 27, so substituting this value into the equation, we have 27 + 9 = 36. Therefore, the number of chairs in the seventh and eighth rows is 36.
Continuing this pattern, we can see that for every second pair of rows, the number of chairs increases by 3. So, the number of chairs in the ninth and tenth rows would be 36 + 3 = 39.
Now let's find the number of chairs in the last two rows, y = 114. Substituting the values we know, y = 39 + 3 = 42.
From the information given, we have that the last two rows each have 114 chairs. Therefore, the number of rows in total can be found by dividing 114 by the number of chairs in the last two rows: 114 / 42 = 2.7.
However, we cannot have a fraction of a row since rows are whole units. Therefore, we need to consider the ceiling or rounding up when converting to a whole number.
Therefore, the answer is that there are a total of 3 rows of chairs in the auditorium.
I hope this explanation helps clarify the solution to the problem. Let me know if you have any further questions!