An auditorium has 20 seats in the 1st row, 24 seats on the second row, 28 seats on the third row, and so on up to the 20th row.. How many seats are there in the last row? How many seats are there in the auditorium?

The given scenario can be modeled as an arithmetic sequence where the number of seats in each row of the auditorium is the corresponding term in the sequence:

20, 24, 28, ...

General form of an arithmetic sequence:

[tex]\boxed{a_n=a+(n-1)d}[/tex]

where:

[tex]a_n[/tex] is the nth term.
a is the first term.
d is the common difference between terms.

For the given sequence 20, 24, 28, ...

a = 20
d = 24 - 20 = 4

Therefore, the equation for the number of seats in the nth row is:

[tex]\boxed{a_n=20+(n-1)4}[/tex]

To find the number of seats in the 20th row, substitute n = 20 into the found formula:

[tex]\begin{aligned}\implies a_{20}&=20+(20-1)4\\& = 20+(19)4\\& = 20+76\\& = 96\end{aligned}[/tex]

Therefore, there are 96 seats in the last row.

Sum of the first n terms of an arithmetic series:

[tex]\boxed{S_n=\dfrac{n}{2}(a+a_n)}[/tex]

To find the total number of seats in the auditorium, substitute n = 20, a = 20 and a₂₀ = 96 into the formula:

[tex]\begin{aligned}\implies S_{20} & = \dfrac{20}{2}(20+96)\\& = 10(116)\\& = 1160\end{aligned}[/tex]

Therefore, there are a total of 1160 seats in the auditorium.

An auditorium has 20 seats in the 1st row, 24 seats on the second row, 28 seats on the third row, and so on up to the 20th row.. How many seats are there in the last row? How many seats are there in the auditorium?​

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An auditorium has 20 seats in the 1st row, 24 seats on the second row, 28 seats on the third row, and so on up to the 20th row.. How many seats are there in the last row? How many seats are there in the auditorium?