Series and sequence

Question

Kevin runs a lawn mowing service. He currently has 14 clients. He has gained 2 new clients at the beginning of each of the past three years. Each year he mows each client’s lawn an average of 15 times. Starting now, Kevin continues to gain 2 clients each year and if he charges $30 per lawn after how many years will he earn a total of $51,300?

Answer 1

8, 10, 12, 14,... x30x15

450x8, 450x10,450x12, 450x14,...

3600, 4500, 5400, 6300,7200,...

when does he reach 51,300

1st term = a1=3600, common difference = d=900, an arithmetic sequence

sn=(a1+an)(n//2)= sum of arithmetic series, n= number of terms

51,300= (3600+an)(n//2)

nan= 102600-3600n, an=a1+d(n-1)=3600+900(n-1)= 2700+900n

n(2700+900n)=2700n+900n^2= 102600-3600n

900n^2 +6300n-102600=0

n^2 +7n -114=0

n-=-7/2+.5sqr(505)

= about 11.2-3.5=7.7

n-3= 4.7 years from now

3600+4500+5400+6300+7200+

8100+9000= $45,000 in 4 years from now

in 5 years = $ 54,000

4.5 years $49,500

4.6 $50,400

4.7 $51,300

$51,300 in 4.7 years from now

4 years 256 days

= 4 yrs 8 mos 12 days

reaches $51,300