The front row of a stadium has 25 seats. Each of the other rows has two more seats than the row in front of it. how many seats are there altogether in the first 20 rows?

i know how to do this in the "long" way but how do you do this is a "shorter" way by finding the pattern? Explain please!!

The seat count forms an arithmetic sequence

25,27,29...

where a = 25, d=2

The sum of the 1st 20 terms is thus
S20 = 20/2 (2*25 + 19*2) = 880

what equation is that?

S= n/2 (a1 + an)

Ah, the joy of patterns! Let's see if we can find a shortcut.

We know that the front row has 25 seats. Now, let's look at the pattern: each row has two more seats than the row in front of it.

If we take a closer look, we'll notice that there are 2 seats added to each row compared to the previous one. So, for the second row, we have 25 + 2 = 27 seats. For the third row, it's 27 + 2 = 29 seats.

Now, let's see if we can find a pattern in the number of seats in each row.

If we keep adding 2 seats to the previous row, we can write it as follows:

Row 1: 25 seats
Row 2: 25 + 2 = 27 seats
Row 3: 27 + 2 = 29 seats

As you can see, the number of seats in each row is increasing by 2.

So, we can use this pattern to find the number of seats in any row. For the Nth row, we just need to take 25 (the number of seats in the first row) and add 2 times the row number (since each row has 2 more seats than the previous row).

In formula form:

Number of seats in the Nth row = 25 + 2N

To find the total number of seats in the first 20 rows, we can substitute N with 1, 2, 3, ... 20 and add them up:

Total number of seats = (25 + 2 * 1) + (25 + 2 * 2) + (25 + 2 * 3) + ... + (25 + 2 * 20)

If we simplify the equation, we get:

Total number of seats = 25 * 20 + 2 * (1 + 2 + 3 + ... + 20)

Now, to calculate the total number of seats, we need to find the sum of the numbers from 1 to 20. Luckily, there is a formula for that:

Sum of numbers from 1 to N = (N * (N + 1)) / 2

So, let's calculate the sum from 1 to 20:

Sum of numbers from 1 to 20 = (20 * 21) / 2 = 210

Now, substituting this into our equation:

Total number of seats = 25 * 20 + 2 * 210 = 500 + 420 = 920 seats.

Voila! There are a grand total of 920 seats in the first 20 rows of the stadium. I hope the pattern helped make it a little less tedious for you!

To find the total number of seats in the first 20 rows, we can use a shorter method by observing the pattern in the number of seats in each row.

We start by looking at the first row, which has 25 seats.

Then, we can observe that each subsequent row has two more seats than the row in front of it. So, the second row will have 25 + 2 = 27 seats, the third row will have 27 + 2 = 29 seats, and so on.

To find the number of seats in the 20th row, we can use the formula:

Number of seats in the nth row = Number of seats in the first row + (n-1) * difference

In this case, the difference between each row is 2. So, substituting the values into the formula, we have:

Number of seats in the 20th row = 25 + (20 - 1) * 2
= 25 + 19 * 2
= 25 + 38
= 63

Therefore, the 20th row will have 63 seats.

To find the total number of seats in the first 20 rows, we can use the formula for the sum of an arithmetic series:

Sum = (Number of terms / 2) * (First term + Last term)

In this case, the number of terms is 20, the first term is 25, and the last term is 63.

Substituting these values into the formula, we have:

Sum = (20 / 2) * (25 + 63)
= 10 * 88
= 880

Therefore, there are a total of 880 seats in the first 20 rows of the stadium.