Create an explicit equation for the recursively defined sequence
t(1)=3, t(n+1)=5t(n)
Once you have created the equation explain how you did so
The given sequence is defined recursively, meaning each term is defined in terms of the previous term. In this case, the first term is defined as t(1) = 3.
To find an explicit equation for the sequence, we need to express each term in terms of its index n.
From the given recursive definition, we can see that each term is obtained by multiplying the previous term by 5. So, we have:
t(2) = 5 * t(1)
t(3) = 5 * t(2) = 5 * (5 * t(1)) = 5^2 * t(1)
t(4) = 5 * t(3) = 5 * (5^2 * t(1)) = 5^3 * t(1)
and so on...
From this pattern, we can generalize the equation for any term t(n) in terms of n and t(1):
t(n) = 5^(n-1) * t(1)
Substituting t(1) = 3 into this equation, we get the explicit equation for the recursively defined sequence:
t(n) = 3 * 5^(n-1)