A geometric sequence is defined by the general term tn = 75(5n), where n ¡ÊN and n ¡Ý 1. What is the recursive formula of the sequence?
Apparently you mean
tn = 75^(5n) = (75^5)^n
tn+1 = tn * 75^5
To find the recursive formula of a geometric sequence, we need to express each term in the sequence using the previous term(s).
In this case, the general term given is tn = 75(5n).
The first term of the sequence can be found by substituting n = 1 into the general term:
t1 = 75(5(1)) = 375
Now, let's find the second term of the sequence. Substituting n = 2 into the general term:
t2 = 75(5(2)) = 750
From these two terms, we can observe that the second term t2 is obtained by multiplying the first term t1 by 2 (5(2) = 2):
t2 = 2 * t1
Now, to find the third term, we substitute n = 3:
t3 = 75(5(3)) = 1125
From the second and third terms, we can observe that the third term t3 is obtained by multiplying the second term t2 by 1.5 (5(3) / 5(2) = 1.5):
t3 = 1.5 * t2
We can continue this pattern, observing that each term is obtained by multiplying the previous term by 5 / 2.
Therefore, the recursive formula for the geometric sequence is:
tn = (5/2) * tn-1
This means that each term is equal to 5/2 times the previous term.