How many seats are in the last row? How many seats are in the theater?

Question

The seats in a local theater are arranged so that there are 64 seats in the first row, 62 seats in the second row, 60 seats in the third row, and so on for 24 rows altogether . How many seats are in the last row? How many seats are in the theater?

Answer 1

you're subtracting "2" 23 times

64-46=18 seats

here's another way to get the sum

S=(n/2)(2a+[n-1]d)

n=# of terms, a=first term, d=common difference

S=(24/2)(2*64+[24-1]*[-2])

S=12(128+23*[-2])

S=12(128-46)

S=12*82

S=984 seats

Answer 2

Row 1 has 64 seats

Row 2 has 62 seats

Row 3 has 60 seats

Row 4 has 58 seats ... and so on.

Rewrite this as

Row 1 has 66 - 2(1) seats

Row 2 has 66 - 2(2) seats

Row 3 has 66 - 2(3) seats

Row 4 has 66 - 2(4) seats

Following this pattern, row "n" has 66 - 2("n") seats, where n is the row number. Since the last row is row 24, there are 66 - 2(24) = 18 seats in the last row.

To calculate the total number of seats, we need to calculate the sum: 64 + 62 + 60 + ... + 22 + 20 + 18, because the number of seats decrease by 2 with each row, and the last row has 18 seats.

There are many ways you can calculate this sum. You could brute force calculate this sum, but that would waste a lot of time. Here's a faster method:

First, you should memorize that 1+2+3+....+n = n(n+1)/2

Note that 64 + 62 + 60 ... + 22 + 20 + 18 = 2( 32 + 31 + 30 + ... 11 + 10 + 9 ), and

32 + 31 + 30 + ... + 11 + 10 + 9 = (32 + 31 + 30 + ... + 3+ 2+ 1) - (8 + 7 + 6 +5 +4 + 3 +2 + 1).

So using our formula above, 32 + 31 + 30 + ... + 11 + 10 + 9 = 32(33)/2 - 8(9)/2.

So 2( 32 + 31 + 30 + ... 11 + 10 + 9 ) = 32(33) - 8(9) = 984.

So there are 984 seats in the theater.