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In an amphitheater, there are 20 seats in the front row, 23 seats in the second row, 26 seats in the third row, and so on, in an arithmetic sequenc A) Write an explicit formula to find an. The number of seats in the nº row.


B) The last row in the amphitheater has 83 seats in it. How many rows a seats are there?


C) Write an expression using sigma notation that represents the total number of seats in the amphitheater.


D) Evaluate this expression. How many total seats are there?

Respuesta :

facundo3141592

For the given arithmetic sequence we have:

  • A) [tex]A_n = 20 + (n - 1)*3[/tex]
  • B) 22 rows.
  • C) [tex]\sum_{n = 1\ to\ 22} (20 + (n - 1)*3)[/tex]
  • D) There are 1,133 seats in total.

How to write the formula for the sequence?

We know that we have 20 seats on the front row, and then we add 3 more on each row, so the formula will be:

[tex]A_n = 20 + (n - 1)*3[/tex]

This gives the number of seats on the n-th row.

B) The last row has 83 seats, then we need to solve:

[tex]A_n = 20 + (n - 1)*3 = 83\\\\(n - 1)*3 = 83 - 20 = 63\\\\n - 1 = 63/3 = 21\\\\n = 21 + 1 = 22[/tex]

So we conclude that there are 22 rows.

C) The total number of seats will be:

[tex]\sum_{n = 1\ to\ 22} (20 + (n - 1)*3)[/tex]

D) Instead of using the sum above (we should add 22 terms) we use the general formula for the sum of N terms in an arithmetic sequence:

[tex]S_N = (N/2)*(2A_1 + (N - 1)*d)[/tex]

In this case, we replace N by 22, and d is the common difference of the sequence, which we know is equal to 3, so we wil have:

[tex]S_{22} = (22/2)*(2*20+ (22 - 1)*3) = 11*(40 + 21*3) = 1,133[/tex]

There are 1,133 seats in total.

If you want to learn more about sequences, you can read:

https://brainly.com/question/7882626

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