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

Answer:

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

600

Step-by-step 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:

$a_5=1.5(1.5a_3-100)-100\\\\a_5=(1.5)^2a_3-100\times (1.5)-100$

Similarly,

$a_3=1.5a_2-100$

so,

$a_5=(1.5)^2\times (1.5a_2-100)-100\times (1.5)-100\\\\a_5=(1.5)^3a_2-(1.5)^2\times 100-(1.5)\times 100-100$

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

$a_5=(1.5)^4a_1-(1.5)^3\times 100-(1.5)^2\times 100-(1.5)\times 100-100$

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

$2225-(1.5)^4a_1-337.5-225-150-100\\\\2225=5.0625a_1-812.5\\\\2225+812.5=5.0625a_1\\\\3037.5=5.0625a_1\\\\a_1=\dfrac{3037.5}{5.0625}\\\\a_1=600$

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

600