How many rows are in the corner section of a stadium containing 2040 seats if the first row has

How many rows are in the corner section of a stadium...

Key Concepts

Quadratic Equation

When solving for the number of terms in an arithmetic sequence given the total sum, the resulting equation can often be reduced to a quadratic form. Solving quadratic equations, typically using the quadratic formula, enables finding the number of terms that satisfy the condition set by the sum.

Arithmetic Series Sum Formula

The sum of an arithmetic series is calculated by multiplying the number of terms by the average of the first and last terms. This formula, S = n/2 * (first term + last term) or S = n/2 * (2a1 + (n-1)d), allows for efficient computation of the total of a sequence with a linear pattern.

Arithmetic Sequence

An arithmetic sequence is a series of numbers where each term after the first is found by adding a constant difference to the previous term. This concept is crucial for understanding problems where there is a regular increase or decrease between terms.

Transcript

00:01 So we're being asked to find how many rows are in the corner section of a stadium that has 2 ,040 seats, if the first row is 10 seats, and each successive row has four additional seats.

00:10 Well, let's think about it.

00:11 There's a total of 240 seats.

00:14 That's going to be our s of n value.

00:17 Now, we're told the first row has 10 seats, so a sub 1 is equal to 10, and we're told that each successive row has four additional seats.

00:25 Well, that's our common difference, so d is equal to four.

00:28 And if we're trying to figure out how many rows there will be, that means we're trying to find n.

00:33 So to do this, we're going to use the formula, s of n is equal to n divided by two times the quantity of 2 times a sub 1 plus n minus 1 times d.

00:48 So let's substitute in the values we have.

00:50 So we're going to have 2040 is equal to n divided by 2 times the quantity of 2 times 10 plus n minus 1 times 4.

01:02 So to solve, i'm first going to multiply both sides of our equation by 2.

01:07 So 2 times 2 ,040 is 4 ,080.

01:13 Then i'm going to bring my n down.

01:15 We have 2 times 10, which is 20.

01:18 And then we're going to distribute the 4.

01:19 So we're going to add positive 4 n minus 4.

01:23 Well, 20 minus 4 is 16.

01:25 So we'll have 4 ,080 is equal to n times the quantity of 16 plus 4 .4.

01:33 Now, let's distribute our n, so we'll get 4 ,080 is equal to 16n plus 4n squared.

01:42 Now, as you can see, we have a quadratic equation, so we're going to set it equal to zero.