In January, Desmond Begins Saving $25 Per Month For A Used Car. Each Month, He Is Able To Save $5 More Than The Previous Month. After How Many Months

In january, desmond begins saving $25 per month for a used car. Each month, he is able to save $5 more than the previous month. After how many months will desmond first have enough money to purchase a $3,000 used car

  Take note that since increase is a fixed amount then this is an arithmetic sequence
well find n which is the number of months he needed to purchase the car

Given are:
a_n \: = 25 \\ d = 5 \\ s_n = 3000
Sum of arithmetic sequence is given by the formula
s_n = \frac{n}{2} (2a_1 + (n - 1)d)
Substitute the given
3000 = \frac{n}{2} (2(25) + (n - 1)(5)) \\ 3000(2) = n(50 + 5n - 5) \\ 6000 = n(5n + 45) \\ 6000 = {5n}^{2} + 45n \\ divide \: everything \: by \: 5 \\ 1200 = {n}^{2} + 9n \\ use \: quadratic \: formula \: to \: find \: n \\ n = 30.43
Therefore Desmond will enough enough money by the 31st month

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