Answer to Question #265685 in Algebra for Wekwek

Question

3. The number of seats in the first row of a theatre has 14 seats. Suppose that each row after the first had 2 additional seats. The number of seats in each row forms an arithmetic sequence.

a) Find the number of seats in the 6th row. [2 marks]

b) Find the total number of seats in the first 10 rows. [3 marks]

c) Given the total number of seats in the first n rows is 660. Show

that n2 13n  660 . Hence, find the value of n when the total number of seats is 660. [6 marks]

Answer 1

The number of seats in the first row of a theatre has 14 seats. Suppose that each row after the first had 2 additional seats. The number of seats in each row forms an arithmetic sequence.

Let $a_n$ be the number of seats in the $n$-th row. Then $a_n=14+2(n-1).$

a) It follows that the number $a_6$ of seats in the $6$-th row is equal to $14+2\cdot 5=24.$

b) Let us use the formula $S_n=\frac{2a_1+d(n-1)}{2}n$ for the sum of $n$ terms of arithmetic sequence. The total number of seats in the first 10 rows is equal to $S_{10}=\frac{2\cdot 14+2(10-1)}{2}10=230.$

c) Let the total number of seats in the first $n$ rows is $660.$ Then we get that$660=\frac{2\cdot 14+2(n-1)}{2}n= (14+(n-1))n=(n+13)n=n^2+13n.$ It follows that $n^2+13n-660=0.$ The last equation is equivalent to $(n-20)(n+33)=0,$ and thus has the roots $n_1=20$ and $n_2=-33.$ Since the number of seats must be positive, we conclude that the value of $n$ is equal to $20.$