The first 3 terms in a geometric sequence are 1.1, 1.65, and 2.475.
Part A: What is the sum of the first 6 terms of the sequence? Show your work.
Part B: How does the sum of the series differ from the 6th term? Explain.
Part C: If the nth term in this sequence represents the total amount of protein a cell has produced in n hours, what would you use to find the amount of protein produced by 4 hours: the 4th term of the sequence, or the sum of the first 4 terms? Explain how you know.
r=1.65/1.1=1.5=22.8594
I have the same question on my exam too!!!
s = a(1-r^6)/(1-r)
= 1.1(1-1.5^6)/(1-1.5)
= 22.8594
correct
s-T6 = 22.8594 - 1.1*1.5^5 = 14.5063
Ummm. the 6th term is the 6th term.
The sum is gotten by adding up all 6 terms.
Use the nth term. It represents the amount produced in n hours. The sum here might represent the total amount in n cells, where each cell is started in successive hours.
Part A: To find the sum of the first 6 terms of a geometric sequence, we can use the formula:
Sn = a * (r^n - 1) / (r - 1)
where Sn represents the sum of the first n terms, a is the first term of the sequence, r is the common ratio, and n is the number of terms.
In this case, the first term (a) is 1.1 and the common ratio (r) can be found by dividing any term by its previous term. For example, r = 1.65 / 1.1 = 1.5.
So we have a = 1.1, r = 1.5, and n = 6.
Plugging these values into the formula, we get:
S6 = 1.1 * (1.5^6 - 1) / (1.5 - 1)
Calculating this expression gives us:
S6 ≈ 1.1 * (7.59375 - 1) / 0.5 ≈ 1.1 * (6.59375) / 0.5 ≈ 7.2546875
Therefore, the sum of the first 6 terms of the sequence is approximately 7.2546875.
Part B: The sum of the series, S6, represents the cumulative total of all the terms up to the 6th term. It takes into account the values of all the terms, adding them up to give the sum. On the other hand, the 6th term is simply the value of the 6th term in the sequence.
So, the difference between the sum of the series and the 6th term is that the sum includes all the terms up to the 6th term, while the 6th term is just a single value in the sequence.
Part C: If the nth term represents the total amount of protein produced by a cell in n hours, we can use the nth term of the sequence to find the amount of protein produced by any specific hour.
In this case, the nth term can be calculated using the formula for the general term of a geometric sequence:
an = a * r^(n-1)
where an is the nth term, a is the first term of the sequence, r is the common ratio, and n is the term number.
To find the amount of protein produced by 4 hours, we need to find the 4th term of the sequence. Using the given values, we have a = 1.1, r = 1.5, and n = 4.
Plugging these values into the formula, we get:
a4 = 1.1 * 1.5^(4-1) = 1.1 * 1.5^3 ≈ 1.1 * 3.375 ≈ 3.7125
Therefore, the amount of protein produced by 4 hours is approximately 3.7125.
We do not need to use the sum of the first 4 terms in this case, as the nth term formula directly gives us the value of the specific term we are interested in.