A daylily farm sells a portion of their daylilies and allows a portion to grow and divide. The recursive formula an=1.5(an-1)-100 represents the number of daylilies, a, after n years. After the fifth year, the farmers estimate they have 2225 daylilies. How many daylilies were on the farm after the first year?

Given : A daylily farm sells a portion of their daylilies and allows a portion to grow and divide. The recursive formula [tex]a_n=1.5(a_{n-1})-100[/tex] represents the number of daylilies, a, after n years. After the fifth year, the farmers estimate they have 2225 daylilies.

To find : How many daylilies were on the farm after the first year?

Solution :

The recursive formula is [tex]a_n=1.5(a_{n-1})-100[/tex]

We have given after the fifth years number of daylilies, [tex]a_5=2225[/tex]

Put n=5 in the formula we get,

[tex]a_5=1.5(a_{5-1})-100[/tex]

[tex]2225=1.5(a_{4})-100[/tex]

[tex]2325=1.5(a_{4})[/tex]

[tex]\frac{2325}{1.5}=a_4[/tex]

[tex]a_4=1550[/tex]

Now, put n=4 in the formula,

[tex]a_4=1.5(a_{4-1})-100[/tex]

[tex]1550=1.5(a_{3})-100[/tex]

[tex]1650=1.5(a_{3})[/tex]

[tex]\frac{1650}{1.5}=a_3[/tex]

[tex]a_3=1100[/tex]

Now, put n=3 in the formula,

[tex]a_3=1.5(a_{3-1})-100[/tex]

[tex]1100=1.5(a_{2})-100[/tex]

[tex]1200=1.5(a_{2})[/tex]

[tex]\frac{1200}{1.5}=a_2[/tex]

[tex]a_2=800[/tex]

Now, put n=2 in the formula,

[tex]a_2=1.5(a_{2-1})-100[/tex]

[tex]800=1.5(a_{1})-100[/tex]

[tex]900=1.5(a_{1})[/tex]

[tex]\frac{900}{1.5}=a_1[/tex]

[tex]a_1=600[/tex]

Which means after first year is [tex]a_1=600[/tex]

Therefore, 600 daylilies were on the farm after the first year.