An auditorium has 30 rows with 10 seats in the first row, 12 in the second row, 14 in the third row, and so forth. How many seats are in the auditorium

My answer is 1170 but the way I figured out the problem was by listing numbers 1-30. the problem state that the 1st row had 10, 2nd row had 12, and 3rd row had 14 and so forth. So i basically did the same method till i got to 30 and then add up all the numbers which gave me the answer of 1170. Does anybody know a shorter formula i can use?

sum of series, n = 1 to 30

10 + (10+2) + (10+2)+2 etc to 10+29(2)
so this last term is 68
so
Sum = 10 + 12 + 14 ....+64 + 66 + 68
Sum = 68 + 66 + 64 ....+14 + 12 + 10
so
2Sum = 78 + 78 + 78 .... +78 + 78 +78
thirty of those 78s is twice the sum
Sum = 30 *78/2
= 1170

OR LOOK UP ARITHMETIC SERIES in your book or Google it :)

Well, that's certainly a thorough way to approach the problem! But let me see if I can lighten the load for you with a shorter formula:

To find the number of seats, you can use an arithmetic sequence formula. The number of seats in each row follows a pattern: 10, 12, 14, and so on. We can see that this sequence starts at 10 and increases by 2 each time.

Now, to calculate the number of seats in the auditorium, you can use the formula for the sum of an arithmetic sequence:

Sum = (n/2) x (first term + last term)

In this case, the first term is 10, the last term can be found using the arithmetic sequence formula:

a_n = a_1 + (n-1)d

where a_n is the nth term, a_1 is the first term, n is the number of terms, and d is the common difference.

So, let's plug in the values:

a_n = 10 + (30-1) x 2
= 10 + 58
= 68

The last term is 68.

Now, let's use the sum formula:

Sum = (30/2) x (10 + 68)
= 15 x 78
= 1170

Voila! Our formula also gives us 1170 seats. Looks like you were on the right track all along. Keep up the good work!

Yes, there is a shorter formula you can use to find the total number of seats in the auditorium.

To find the number of seats in each row, we can observe that the pattern is increasing by 2 seats for each row. The first row has 10 seats, the second row has 10 + 2 = 12 seats, the third row has 12 + 2 = 14 seats, and so on.

We can express the number of seats in each row as a mathematical expression using the row number:
Number of seats in a row = 10 + (row number - 1) * 2

To find the total number of seats in the auditorium, we can sum up the number of seats in each row from the first row to the last row (30th row):

Total number of seats = (10 + (1 - 1) * 2) + (10 + (2 - 1) * 2) + (10 + (3 - 1) * 2) + ... + (10 + (30 - 1) * 2)

Simplifying this expression, we get:

Total number of seats = 10 * 30 + 2 * (1 + 2 + 3 + ... + 30)

Using the formula for the sum of consecutive numbers (1 + 2 + 3 + ... + n = n * (n + 1) / 2), we can further simplify:

Total number of seats = 10 * 30 + 2 * (30 * (30 + 1) / 2)

Total number of seats = 300 + 2 * (30 * 31 / 2)

Total number of seats = 300 + 2 * 465

Total number of seats = 300 + 930

Total number of seats = 1230

So, the auditorium has a total of 1230 seats, not 1170 as you initially calculated.

thanks Damon

(n/2)(a1+an)

or
(n/2)[2a1 + (n-1)d]

An auditorium has 30 rows with 10 seats in the first row, 12 in the second row, 14 in the third row, and so forth. How many seats are in the auditorium

My answer is 36, because there as 10 seats in the first row I added it with 14 and 12 because 14 and 12 is equal to 26 if you add a ten thats 36. I'm not the smartest but I think my answer makes sense.