Question

4. A large university amphitheatre has semi-circle seating with
30 seats in the first row, 32 seats in the second row, 34 seats in
the third row, and so on. The theatre has 50 rows.
a) Develop a formula to calculate the number of seats in
the first n rows.
b) How many seats are there in the first 12 rows?
c) How many seats are available in total in the whole
amphitheatre?

Answer 1

Answer:

a) Sum = 28n + n² + n

b) 492 seats

c) 3950 seats

Step-by-step explanation:

The given data forms arithmetic series.

30, 32, 34,.........

Total rows = n = 50.

a = first term = 30

Common difference = d = second term - first term

                                       = 32 -30

                                        = 2

a) Sum =  (28 + 2) + (28 + 4) +.................+ 28 + 2n

           = 28n + (2 + 4 + ...... +2n)      

Sum of first 'n' even numbers = n*(n+1)

          = 28n + n *(n + 1)

          = 28n + n² + n

b)  To find the number seats in the first 12 rows, use the formula,

          $\boxed{\bf S_n =28n +n^2 + n}$

           $\sf S_{12} = 28*12 +12^2 + 12$

                = 336 + 144 + 12

                = 492

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c)

     $\sf S_{50} =28*50 + 50^2 + 50$

           = 1400 + 2500 +50

           =3950

Total seat available in the whole amphitheatre = 3950 seats