A geometric sequence is defined by the general term t^n = 75(5^n), where n ∈N and n ≥ 1. What is the recursive formula of the sequence?
A) t^1 = 75, t^n = 5t^n - 1, where n ∈N and n > 1
B) t^1 = 75, t^n = 75t^n - 1, where n ∈N and n > 1
C) t^1 = 375, t^n = 5t^n - 1, where n ∈N and n > 1
D) t^1 = 375, t^n = 5t^n + 1, where n ∈N and n > 1
Please help! I'm really confused.
note that 5^n multiples by 5 for each increase in n.
So, B is out.
when n=1, 75*5^1 = 375, so that is t^1
C is correct, if written 5^(n-1)
rather than t^n you should write tn or t_n to avoid confusing the subscript with an exponent.
oops. I meant t_(n-1), not t^(n-1)
see what I meant?
thank you steve
To find the recursive formula for a geometric sequence, you need to identify the relationship between consecutive terms in the sequence.
In this case, the general term is given by t^n = 75(5^n), where n is a natural number greater than or equal to 1.
By substituting values for n, we can determine the first few terms of the sequence:
When n = 1, t^1 = 75(5^1) = 375
When n = 2, t^2 = 75(5^2) = 1875
When n = 3, t^3 = 75(5^3) = 9375
And so on...
Looking at the first few terms, we can determine the relationship between consecutive terms. In this case, each term is obtained by multiplying the previous term by 5. Therefore, the recursive formula can be written as:
t^1 = 375 (the first term)
t^n = 5t^n-1 (where n is a natural number greater than 1)
Comparing this recursive formula to the given choices, we can see that the correct answer is:
A) t^1 = 75, t^n = 5t^n-1, where n is a natural number greater than 1