Each year for 4 years, a farmer increased the number of trees in a certain orchard by of the number of trees in the orchard the preceding year. If all of the trees thrived and there were 6,250 trees in the orchard at the end of the 4-year period, how many trees were in the orchard at the beginning of the 4-year period?

Let’s say that x is number of trees at the begging of the first year, we know that for four years the number of trees were incised by 1/4 of the number of trees of the preceding year, so at the end of the first year the number of trees was[tex]x+\frac{1}{4} x=\frac{5}{4} x[/tex], and for the next three years we have that

Start End

Second year [tex]\frac{5}{4}x[/tex] -------------- [tex]\frac{5}{4}x+\frac{1}{4}(\frac{5}{4}x) =\frac{5}{4}x+ \frac{5}{16}x=\frac{25}{16}x=(\frac{5}{4} )^{2}x[/tex]

Third year [tex](\frac{5}{4} )^{2}x[/tex]-------------[tex](\frac{5}{4})^{2}x+\frac{1}{4}((\frac{5}{4})^{2}x) =(\frac{5}{4})^{2}x+\frac{5^{2} }{4^{3} } x=(\frac{5}{4})^{3}x[/tex]

Fourth year [tex](\frac{5}{4})^{3}x[/tex]--------------[tex](\frac{5}{4})^{3}x+\frac{1}{4}((\frac{5}{4})^{3}x) =(\frac{5}{4})^{3}x+\frac{5^{3} }{4^{4} } x=(\frac{5}{4})^{4}x.[/tex]

So the formula to calculate the number of trees in the fourth year is

[tex](\frac{5}{4} )^{4} x,[/tex] we know that all of the trees thrived and there were 6250 at the end of 4 year period, then

[tex]6250=(\frac{5}{4} )^{4}x[/tex]⇒[tex]x=\frac{6250*4^{4} }{5^{4} }= \frac{10*5^{4}*4^{4} }{5^{4} }=2560.[/tex]

Therefore the number of trees at the begging of the 4-year period was 2560.