Answer:
Hence, the number of daylilies in the first year were:
600
Step-by-step explanation:
It is given that:
the recursive formula [tex]a_n=1.5(a_{n-1})-100[/tex] 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.
[tex]a_5=2225[/tex]
Also,
[tex]a_5=1.5a_4-100[/tex]
[tex]a_4=1.5a_3-100[/tex]
This means that:
[tex]a_5=1.5(1.5a_3-100)-100\\\\a_5=(1.5)^2a_3-100\times (1.5)-100[/tex]
Similarly,
[tex]a_3=1.5a_2-100[/tex]
so,
[tex]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[/tex]
and so, putting [tex]a_2[/tex] in terms of [tex]a_1[/tex] we get:
[tex]a_5=(1.5)^4a_1-(1.5)^3\times 100-(1.5)^2\times 100-(1.5)\times 100-100[/tex]
Now on putting the value of [tex]a_5[/tex] we find the value of [tex]a_1[/tex]
[tex]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[/tex]
Hence, the number of daylilies in the first year were:
600
Answer:
The answer is 600 (ie, C)
Step-by-step explanation:
