Answer:
Step-by-step explanation:
1). [tex]a_1=\frac{2}{5}[/tex]
[tex]a_n=5a_{n-1}[/tex]
Since, recursive formula for a geometric sequence is given by,
[tex]a_n=r.a_{n-1}[/tex]
Here, r = common ratio
Comparing both, value of r = 5
Explicit formula of the sequence is given by,
[tex]a_n=a_1(r)^{n-1}[/tex]
Therefore, explicit formula will be,
[tex]a_n=\frac{2}{5}(5)^{n-1}[/tex]
2). Given recursive formula is,
[tex]a_n=\frac{1}{2}(\frac{4}{3})^{n-1}[/tex]
By comparing this formula with the standard recursive formula for geometric sequence,
[tex]a_n=a_1(r)^{n-1}[/tex]
[tex]a_1=\frac{1}{2}[/tex]
[tex]r=\frac{4}{3}[/tex]
Therefore, recursive formula for the sequence will be,
[tex]a_1=\frac{1}{2}[/tex]
[tex]a_n=a_{n-1}(\frac{4}{3})[/tex]
3). Let the explicit formula is,
[tex]a_n=a_1(r)^{n-1}[/tex]
For 2 years of investment, [tex]a_n=550000[/tex], [tex]a_1=500000[/tex] and n = 2
550000 = [tex]500000(r)^{2-1}[/tex]
[tex]r=\frac{550000}{500000}[/tex]
r = 1.1
Therefore, explicit formula will be.
[tex]a_n=500000(1.1)^{n-1}[/tex]
Option (3) is the answer.
4). Explicit formula for a geometric sequence is,
[tex]a_n=a_1(r)^{n-1}[/tex]
Here [tex]a_1[/tex] = First term
r = common ratio
from the given sequence,
[tex]a_1[/tex] = 9
r = [tex]\frac{a_2}{a_1}[/tex]
r = [tex]\frac{6}{9}=\frac{2}{3}[/tex]
Explicit formula will be,
[tex]a_n=9(\frac{2}{3})^{n-1}[/tex]
5). Recursive formula of a geometric sequence is given by,
[tex]a_1[/tex] = First term of the sequence
[tex]a_n=a_{n-1}(r)[/tex]
From the given sequence,
[tex]a_1=4[/tex]
r = [tex]\frac{-16}{4}=-4[/tex]
Therefore, recursive formula will be,
[tex]a_1=4[/tex]
[tex]a_n=a_{n-1}(-4)[/tex] = [tex]-4a_{n-1}[/tex]
