Which of the following three numbers, inserted between 16 and .0625 form a series of 5 consecutive terms in a geometric sequence? Select all that apply
nvm got it
.0625 = 1/16
so, 16r^4 = 1/16
r^4 = 1/256 = 1/2^8
r = 1/2^2 = 1/4
So, the 5 terms are
16, 4, 1, 1/4, 1/16
To determine which numbers form a series of 5 consecutive terms in a geometric sequence, we can use the formula for a geometric sequence:
\(a_n = a_1 \cdot r^{(n-1)}\)
In this formula, \(a_n\) represents the nth term of the sequence, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term in the sequence.
In this case, we are given the first two terms: 16 and 0.0625. Let's consider the three numbers given and check if they can be the third term in the sequence.
1) 4
Using the formula, we can calculate the third term:
\(a_3 = 16 \cdot r^{(3-1)}\)
\(4 = 16 \cdot r^{2}\)
Dividing both sides by 16:
\(0.25 = r^{2}\)
Taking the square root of both sides:
\(r = 0.5\) or \(r = -0.5\)
2) 0.25
Using the formula, we can calculate the third term:
\(a_3 = 16 \cdot r^{(3-1)}\)
\(0.25 = 16 \cdot r^{2}\)
Dividing both sides by 16:
\(0.015625 = r^{2}\)
Taking the square root of both sides:
\(r = 0.125\) or \(r = -0.125\)
3) 0.5
Using the formula, we can calculate the third term:
\(a_3 = 16 \cdot r^{(3-1)}\)
\(0.5 = 16 \cdot r^{2}\)
Dividing both sides by 16:
\(0.03125 = r^{2}\)
Taking the square root of both sides:
\(r = 0.177\) or \(r \approx -0.177\)
From these calculations, we can see that only the number 0.25 can be the third term in a geometric sequence with the given first two terms of 16 and 0.0625.
Therefore, the correct answer is: 0.25