Answer:
Explanation:
A) In a geometric sequence, t_(1)=625 and the common ratio is r=(1)/(5). For what value of n is t_(n)=(1)/(25)?
Step 1: Write the formula for the nth term
The formula for the nth term of a geometric sequence is t_n = t_1 * r^(n-1), where t_1 is the first term and r is the common ratio.
Step 2: Substitute the given values into the formula
t_n = 625 * (1/5)^(n-1)
Step 3: Solve for n when t_n = (1/25)
(1/25) = 625 * (1/5)^(n-1)
(1/25) / 625 = (1/5)^(n-1)
(1/5)^(n-1) = (1/25) / 625
(n-1) = log base (1/5) of ((1/25) / 625)
Step 4: Solve for n
n = 1 + log base (1/5) of ((1/25) / 625)
The value of n is the number of terms that have passed when t_n = (1/25).
B) (2)/(5),(3)/(6),(4)/(7),(5)/(8),...
Step 1: Determine the type of sequence
A sequence is an arithmetic sequence if the difference between any two consecutive terms is constant. Since this sequence does not have a constant difference, it is not an arithmetic sequence.
A sequence is a geometric sequence if the ratio between any two consecutive terms is constant. We can see that the ratio between any two consecutive terms is constant, so this sequence is a geometric sequence.
Step 2: Find the common ratio
To find the common ratio, we can divide any two consecutive terms. For example, the common ratio is (3/6) / (2/5) = (3/2) / (6/5) = 5/6.
Step 3: Write the formula for the nth term
The formula for the nth term of a geometric sequence is t_n = t_1 * r^(n-1), where t_1 is the first term and r is the common ratio.
Step 4: Substitute the given values into the formula
To use this formula, we need to know the value of t_1. We are given t_1 = (2/5), so:
t_n = (2/5) * (5/6)^(n-1)
This is the formula for the nth term of the sequence (2)/(5),(3)/(6),(4)/(7),(5)/(8),...