Create an explicit equation for the recursively defined sequence
t(1)=3, t(n+1)=5t(n)
The recursive equation for the sequence is defined as follows:
t(1) = 3,
t(n+1) = 5t(n).
To find the explicit equation for this recursively defined sequence, we can observe the pattern and find a formula that directly relates a term (t(n)) to its position (n).
Let's examine the first few terms of the sequence:
t(1) = 3,
t(2) = 5t(1) = 5 * 3 = 15,
t(3) = 5t(2) = 5 * 15 = 75,
t(4) = 5t(3) = 5 * 75 = 375,
........
From the pattern, we can see that each term is obtained by multiplying the previous term by 5.
Therefore, the explicit equation for this recursively defined sequence is:
t(n) = 3 * 5^(n-1).
So, the explicit equation for the given sequence is t(n) = 3 * 5^(n-1).