I’ve gotten it wrong many times

Question

After beginning a new position with 9 established accounts, a salesman made it his objective to add 3 new accounts every month. His goal was to reach 51 accounts. At this rate, how many months would it take to reach his goal?

Answer 1

Note: " 51 = 9 + (n - 1)3 " is the formula to use to find the n-th element in a series that starts with element a₁.= 3. However, here, a₀ is the end of month=0. Sometimes, in math and engineering, a series begins with a₀.

This problem is easy enough to list the totals through the end of month 14:

Answer 2

Since he already has 9 established accounts, we need to subtract this from 51. 51 - 9 = 42, meaning he needs 42 new accounts to reach his goal of 51 accounts total. If he adds 3 new accounts every month, we want to divide 42 by 3. 42 / 3 = 14. It will take him 14 months to reach his goal.

Another way of thinking about it is as a linear function in algebra 1. The equation would be y = 3x + 9, because he already has 9 accounts (making it our y-intercept) and the rate of change is plus 3 every month (if x is the number of months then we would want to multiply the number of months by 3 to get all the accounts from those months. Setting y = 51 gives us: 51 = 3x + 9. Subtract 9 from both sides gives us: 42 = 3x. Dividing by 3 on both sides gives us: 14 = x.

Answer 3

51 = 9 + (n - 1)3

42 = (n - 1)3

14 = n - 1

15 = n