Arithmetic Series question

Question

'A merchant places $1 in a jackpot on August 1, then draws the name of a regular customer. If the customer is present, he or she wins the $1 in the jackpot. If the customer is not present, the merchant adds $2 to the jackpot on August 2 and draws another name. Each day the merchant adds an amount equal to the day of the month. If the first person to win the jackpot wins $496, on what day of the month was his or her name drawn?'

Answer 1

To quickly find the sum of a set of consecutive numbers, you can add the first and the last and multiply by half the number of values in the list. Based on that, we get the equation:

 

(1 + x)(x/2) = 496     where x is the day of the award.

x² + x = 992             distribute and multiply both sides by 2

x² + x - 992 = 0

(x + 32)(x - 31) = 0    factor and apply the zero product rule

x = -32, 31                 can't have negative days

x = 31