1. Find the value of k so that 2, k^2, 8 form a geometric sequence.
2. Water is leaking from a water tower. On the first day, two gallons were lost. The leak is getting progressively worse and the amount of water lost triples each day. How many gallons would be lost on the 8th day?
1. if GS , then k^2/2 = 8/k^2
k^4 = 16
k^2 = ± 4
but k^2 can't be -4
k^2 = 4
k = ± 2
2. looks like a = 2 , r = 3
find term(8)
Just use your basic formula
1. To find the value of k, we need to determine the common ratio of the sequence. In a geometric sequence, each term is obtained by multiplying the previous term by the common ratio.
To find the common ratio, we can divide any term by the previous term. Let's divide the second term by the first term, (k^2 / 2). Since the second term is obtained by multiplying the first term by the common ratio, we can set up the equation:
k^2 / 2 = 8 / k
Now we can solve for k. Multiply both sides of the equation by k:
k^3 / 2 = 8
Multiply both sides by 2:
k^3 = 16
Taking the cube root of both sides, we find:
k = ∛16
Therefore, the value of k is the cube root of 16.
2. The amount of water lost on each day forms a geometric sequence with a common ratio of 3. On the first day, 2 gallons were lost. To find the amount of water lost on the 8th day, we need to calculate the term of the geometric sequence corresponding to that day.
The formula to find the nth term of a geometric sequence is:
an = a1 * r^(n-1)
where an is the nth term, a1 is the first term, r is the common ratio, and n is the number of terms.
In this case, a1 = 2 (the amount lost on the first day), r = 3 (the common ratio), and n = 8 (the 8th day). Substituting these values into the formula:
a8 = 2 * 3^(8-1)
a8 = 2 * 3^7
Now we can calculate the value of a8:
a8 = 2 * 3^7 = 2 * 2187 = 4374
Therefore, on the 8th day, 4374 gallons would be lost.