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Sequences: Consecutive Integers - Calculating the Sum of Consecutive Integers

Tommy spent six lucky days in Las Vegas. On his first day, he won a net amount of $20, and on each of the following days, the daily net amount he won grew by $$\$d$$. If Tommy won a total net amount of $1,620 during his stay in Las Vegas, how much did he win on the last day?
**Alternative Solution**: Instead, you could solve for $$d$$ by writing out the amount Tommy won on each day and then writing an equation where they add up to $1,620. - 1st day: $$\$20$$ - 2nd day: $$\$20 + \$d$$ - 3rd day: $$\$20 + \$2d$$ - 4th day: $$\$20 + \$3d$$ - 5th day: $$\$20 + \$4d$$ - 6th day: $$\$20 + \$5d$$ Adding those up and setting them equal to $1,620 gives us > $$\$20\cdot 6 + \$15d = \$1{,}620$$ > $$\$120 + \$15d = \$1{,}620$$ > $$\$15d = \$1{,}500$$ > $$d = 100$$. That means that on the sixth day, Tommy made $520.
Incorrect. [[Snippet]]
Incorrect. [[Snippet]]
Incorrect. [[Snippet]]
Incorrect. [[Snippet]]
Correct. [[Snippet]] Since the overall average amount he made is the average of the first and last days, it is > $$\displaystyle \text{Average} = \frac{x+\$20}{2}$$. Putting that into the formula for finding the average, you get > $$\displaystyle \frac{x+\$20}{2} = \frac{\$ 1{,}620}{6}$$ > $$\displaystyle \frac{x+\$20}{2} = \$ 270$$ > $$\displaystyle x+\$20 = \$ 540$$ > $$\displaystyle x = \$ 520$$. Hence, this is the correct answer.
$330
$500
$520
$540
$620
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