For each sequence defined recursively, write the first few terms. Then use the terms to write an explicit equation.
a(1)=17 a(n+1)=a(n)-3
And
a(1) =20. a(n+1)=1/2*a(n)
a(1)=17 a(n+1)=a(n)-3
a(2) = a(1) - 3 = 17-3 = 14
a(3) = a(2) - 3 = 14-3 = 11
decreases by 3 each time, ? - 3 = 17
looks like a(n) = -3n+ 20
a(1) =20. a(n+1)=1/2*a(n)
a(2) = (1/2) a(1) = 10
a(3) = (1/2) a(2) = 5
so 20, 10, 5 <---- a GS with a = 20 r = 1/2
t(n) = ar^(n-1)
t(n) = 20 (1/2)^(n-1)
To find the first few terms of the sequence and write an explicit equation, we'll follow these steps for each sequence:
Sequence 1:
Given the recursive definition:
a(1) = 17
a(n+1) = a(n) - 3
1. Find the first few terms:
a(1) = 17
a(2) = a(1) - 3 = 17 - 3 = 14
a(3) = a(2) - 3 = 14 - 3 = 11
2. Write an explicit equation:
To find an explicit equation, look for a pattern in the terms. We can notice that each term is obtained by subtracting 3 from the previous term. So, the explicit equation for this sequence would be:
a(n) = 20 - 3(n-1)
Therefore, the explicit equation for the first sequence is: a(n) = 17 - 3(n-1).
Sequence 2:
Given the recursive definition:
a(1) = 20
a(n+1) = (1/2) * a(n)
1. Find the first few terms:
a(1) = 20
a(2) = (1/2) * a(1) = (1/2) * 20 = 10
a(3) = (1/2) * a(2) = (1/2) * 10 = 5
2. Write an explicit equation:
To find an explicit equation, we can observe that each term is half of the previous term. So, the explicit equation for this sequence can be written as:
a(n) = 20 * (1/2)^(n-1)
Therefore, the explicit equation for the second sequence is: a(n) = 20 * (1/2)^(n-1).
Now you have the explicit equations for both sequences.