Question

In a theater, there are 15 chairs in the first row, each row has 3 more chairs than the previous row, and there are 15 rows. how many chairs are there in the theater?

Answer 1

The 1st row has 15 chairs.

The 2nd row has 15 + 3 = 18 chairs.

The 3rd row has 18 + 3 = 15 + 2•3 = 21 chairs.

The 4th row has 21 + 3 = 15 + 3•3 = 24 chairs.

The pattern continues, so that the n-th row has 15 + (n - 1)•3 = 3n + 12 chairs.

If there are 15 rows, then the total number of chairs is

`\displaystyle \sum_(n=1)^(15) (3n + 12) = 3 \sum_(n=1)^(15) n + 12 \sum_(n=1)^(15) 1 = 3\cdot\frac{15\cdot16}2 + 12\cdot15 = \boxed{540}`

where we use the formulas

`\displaystyle \sum_(n=1)^N 1 = N`

`\displaystyle \sum_(n=1)^N n = \frac{N(N+1)}2`