The first row of seating in section A of an arena has 7 seats. There are 25 rows in section A, each row containing two more seats than the row preceeding it. How many seats are in section A?
How do you use geometric sequence to solve it?
the sequence is not geometric, but arithmetic.
a=7
d=2
S25 = 25/2 (2*7+24*2)
To use a geometric sequence to solve this problem, we need to find the common ratio between consecutive terms.
In this scenario, the first row has 7 seats. Let's call this "a₁" (the first term). We can also represent the number of seats in each row as "a₂", "a₃", and so on.
Since each row contains two more seats than the row preceding it, the common difference between the terms is 2. So, we can say that a₂ = a₁ + 2, a₃ = a₂ + 2, and so on.
Now, let's calculate the number of seats in section A by summing the terms of the geometric sequence.
The formula for the nth term of a geometric sequence is:
aₙ = a₁ * r^(n-1)
where "r" represents the common ratio.
We know that a₁ = 7, and we need to find the value of a₂ to determine the value of "r".
Since a₂ = a₁ + 2, we can substitute the values:
a₂ = 7 + 2 = 9
Now, we can find the value of "r" using the formula:
a₂ = a₁ * r^(2-1)
9 = 7 * r¹
Simplifying, we get:
9 = 7r
Dividing both sides of the equation by 7, we find that:
r = 9/7
With the common ratio determined, we can use the formula for the sum of a geometric sequence to calculate the total number of seats in section A.
The sum of a geometric sequence is given by the formula:
Sₙ = a₁ * (1 - rⁿ) / (1 - r)
Plugging in the known values, we get:
Sₙ = 7 * (1 - (9/7)^25) / (1 - 9/7)
Evaluating this expression will give you the total number of seats in section A.
To use a geometric sequence to solve this problem, we need to understand the pattern of the number of seats in each row of section A.
Given that the first row has 7 seats, and each row has two more seats than the row preceding it, we can deduce that the number of seats in each row forms a geometric sequence. The common ratio between consecutive terms of this sequence is 2, as each row has two more seats than the previous row.
To find the number of seats in section A, we can sum the terms of this geometric sequence. The formula to calculate the sum of the n terms of a geometric sequence is:
Sn = a * ( r^n - 1) / ( r - 1)
Where:
Sn is the sum of the first n terms,
a is the first term of the sequence,
r is the common ratio.
In this case, the first term (a) is 7, and the common ratio (r) is 2. Since there are 25 rows in section A, we can substitute these values into the formula:
Sn = 7 * ( 2^25 - 1) / ( 2 - 1)
Now, we can compute the value of Sn using a calculator or by simplifying the expression. This will give us the total number of seats in section A.