# Question #4566a

Nov 21, 2017

$285750$ buns

#### Explanation:

First, try to write it as a summation.

We know that on day 1 the bakery made 200 buns and that every subsequent day the bakery made 5 more buns until the last day, on which they made 1695 buns. How many days is that?

$\frac{1695 - 200}{5} = 299$

Since 200 is day 1, 1695 must be day 300. (If you don't understand this, I have a different explain it at the bottom).

Now, we can write a summation. We know we must increment by 5 each time from day 1 to day 300.

${\sum}_{i = 1}^{300} \left(200 + 5 i\right)$

$= {\sum}_{i = 1}^{300} \left(200\right) + {\sum}_{i = 1}^{300} \left(5 i\right)$

The first sum is just adding the number 200 three-hundred times. So $200 \cdot 300 = 60000$.

${\sum}_{i = 1}^{300} \left(5 i\right) = 5 {\sum}_{i = 1}^{300} \left(i\right)$

The second sum is two steps: adding all the numbers from 1 to 300, then multiplying this by 5.

The sum of all numbers from 1 to 300 is $300 \cdot \frac{300 + 1}{2} = 45150$. Multiplying this by 5, we get $225750$.

$225750 + 60000 = 285750$ buns

(side note)
As for that explanation, imagine if we tried to create an equation based on the day. For day 1, we get 200. For day 2, we get 205. Since we know it's linear, we can create a line:

slope: $\frac{205 - 200}{2 - 1} = 5$

point-slope: $y - 200 = 5 \left(x - 1\right)$

slope-intercept: $y = 5 x + 195$

Now, we want $x$ such that $y = 1695$.

$1695 = 5 x + 195$

$5 x = 1500$

$x = 300$