Answer: A. $25.36, $75.99
Step-by-step explanation:
Given recursive formula: [tex]a_n=1.025(a_{n-1})+50[/tex]
Also, [tex]a_6=291.51[/tex]
Put n=6 in the above formula , we get
[tex]a_6=1.025(a_{6-1})+50\\\Rightarrow291.51=1.025(a_5)+50\\\Rightarrow1.025(a_5)=291.51-50\\\Rightarrow1.025(a_5)=241.51\\\Rightarrow(a_5)=235.619512195[/tex]
Similarly, for n=5 in the above formula , we get
[tex]a_5=1.025(a_{5-1})+50\\\Rightarrow235.619512195=1.025(a_4)+50\\\Rightarrow1.025(a_4)=235.619512195-50\\\Rightarrow1.025(a_4)=185.619512195\\\Rightarrow(a_4)=181.09220702[/tex]
for n=4 in the above formula , we get
[tex]a_4=1.025(a_{4-1})+50\\\Rightarrow181.09220702=1.025(a_3)+50\\\Rightarrow1.025(a_3)=181.09220702-50\\\Rightarrow1.025(a_3)=131.09220702\\\Rightarrow(a_3)=127.894836117[/tex]
for n=3 in the above formula , we get
[tex]a_3=1.025(a_{3-1})+50\\\Rightarrow127.894836117=1.025(a_2)+50\\\Rightarrow1.025(a_2)=127.894836117-50\\\Rightarrow1.025(a_2)=77.894836117\\\Rightarrow(a_2)=75.9949620654\approx75.99[/tex]
for n=2 in the above formula , we get
[tex]a_2=1.025(a_{2-1})+50\\\Rightarrow75.99=1.025(a_1)+50\\\Rightarrow1.025(a_1)=75.99-50\\\Rightarrow1.025(a_1)=25.99\\\Rightarrow(a_1)=25.356097561\approx25.36[/tex]
Hence, the first two numbers in the sequence are $25.36 and $75.99.
