Please...I really need help on this question...Thanks.
-Noah is trying to create a sequence for his group. He wants the common ratio to be 4, he wants the 5th term to be 15 and he wants the 8th term to be 900. However, this is NOT possible! Explain to Noah why it is not possible create the sequence he has in mind.
15 is less than 900
so it cannot come later in the sequence if the ratio is greater than 1.
oops.
900/15 is not a power of 4
so you know r = 4
term5 = ar^4 = 15
term8 = ar^7 = 900
divide one by the other: ar^7/(ar^4) = 900/15
r^3 = 60
clearly r ≠ 4
could have worked if term8 = 960
To explain why it is not possible to create the sequence Noah has in mind, we need to understand the concept of a geometric sequence and how the terms in the sequence are related to each other.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant value called the common ratio (r). The general formula for the nth term (Tn) in a geometric sequence is:
Tn = a * r^(n-1)
Where:
- a is the first term in the sequence,
- r is the common ratio,
- n is the position of the term we are trying to find.
Now, considering the information Noah has given us:
- The common ratio (r) is 4.
- The 5th term of the sequence should be 15, which means T5 = 15.
- The 8th term of the sequence should be 900, which means T8 = 900.
Let's use this information along with the formula for the nth term to solve for the first term (a):
T5 = a * r^(5-1) = a * r^4 = 15
T8 = a * r^(8-1) = a * r^7 = 900
Now, we can use the two equations to solve for a and r:
a * r^4 = 15 ...(Equation 1)
a * r^7 = 900 ...(Equation 2)
We can divide Equation 2 by Equation 1 to eliminate a:
(r^7) / (r^4) = 900 / 15
r^3 = 60
Taking the cube root of both sides:
r = ∛60
Now, if we substitute this value of r into Equation 1, we can solve for a:
a * (∛60)^4 = 15
a * (∛60)^4 = 15
a * (∛60^4) = 15
a * (∛60^3 * ∛60) = 15
a * (∛60^3 * 2 * ∛15) = 15
a * (60^(3/2) * ∛15) = 15
As we simplify this equation, we find that there is no whole number value for a that satisfies the equation.
Therefore, it is not possible to create the sequence Noah has in mind with a common ratio of 4, a 5th term of 15, and an 8th term of 900.