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 seriesy 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.
Part A: To find the sum of the first 6 terms of a geometric sequence, you can use the formula:
Sn = a * (r^n - 1) / (r - 1)
Where:
Sn is the sum of the first n terms,
a is the first term of the sequence,
r is the common ratio,
n is the number of terms in the sequence.
In this case, the first term (a) is 1.1, and the common ratio (r) can be found by dividing the second term by the first term: 1.65 / 1.1 = 1.5. Let's substitute these values into the formula:
S6 = 1.1 * (1.5^6 - 1) / (1.5 - 1)
Calculating this expression, we get:
S6 ≈ 1.1 * (7.59375 - 1) / 0.5
S6 ≈ 1.1 * (6.59375) / 0.5
S6 ≈ 1.1 * 13.1875
S6 ≈ 14.50625
Therefore, the sum of the first 6 terms of the sequence is approximately 14.50625.
Part B: The sum of the series differs from the 6th term because the sum includes the cumulative effect of all the previous terms in the sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio. So as you progress through the terms, the value of each term increases. The sum of the series takes into account all the terms up to the specified number, while the 6th term only refers to the value at that particular position in the sequence. The sum is the cumulative result, while the 6th term is an individual value.
Part C: If the nth term in the given sequence represents the total amount of protein produced by a cell in n hours, and you want to find the amount of protein produced in 4 hours, you would use the sum of the first 4 terms. This is because the sum of the first 4 terms represents the cumulative total protein produced up to the 4th hour. On the other hand, the 4th term of the sequence only represents the protein produced at the 4th hour and does not take into account the protein produced in the previous hours. Therefore, to find the total amount of protein produced by the cell after 4 hours, you would need to consider the sum of the first 4 terms.