Question

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. In the fifth year, the farmers estimate they have 2,225 daylilies. How many daylilies were on the farm in the first year?

279

518

600

800





THE ANSWER IS C. 600, I TOOK THE TEST

Answer 1

The answer is 600 (ie, C)

Explanation:

Answer 2

Hence, the number of daylilies in the first year were:

600

Explanation:

It is given that:

the recursive formula
`a_n=1.5(a_(n-1))-100` represents the number of daylilies, a, after n years.

Also we are given that in the fifth year they have 2,225 daylilies.

i.e.

`a_5=2225`

Also,

`a_5=1.5a_4-100`

`a_4=1.5a_3-100`

This means that:

`a_5=1.5(1.5a_3-100)-100\\\\a_5=(1.5)^2a_3-100* (1.5)-100`

Similarly,

`a_3=1.5a_2-100`

so,

`a_5=(1.5)^2* (1.5a_2-100)-100* (1.5)-100\\\\a_5=(1.5)^3a_2-(1.5)^2* 100-(1.5)* 100-100`

and so, putting
`a_2` in terms of
`a_1` we get:

`a_5=(1.5)^4a_1-(1.5)^3* 100-(1.5)^2* 100-(1.5)* 100-100`

Now on putting the value of
`a_5` we find the value of
`a_1`

`2225-(1.5)^4a_1-337.5-225-150-100\\\\2225=5.0625a_1-812.5\\\\2225+812.5=5.0625a_1\\\\3037.5=5.0625a_1\\\\a_1=(3037.5)/(5.0625)\\\\a_1=600`

Hence, the number of daylilies in the first year were:

600