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.

A) r= 1.65/1.1=1.5=22.8594

To find the sum of the first 6 terms of a geometric sequence, there are two commonly used formulas:

Formula 1: Sn = a * (1 - r^n) / (1 - r)
Formula 2: Sn = a * (r^n - 1) / (r - 1)

In both formulas, Sn denotes the sum of the first n terms, a represents the first term in the sequence, and r stands for the common ratio.

Part A: To find the sum of the first 6 terms of the given geometric sequence (1.1, 1.65, 2.475), we need to determine the values of a and r.

The first term (a) is 1.1, and the second term (1.65) is obtained by multiplying the first term by the common ratio (r). Therefore, we can solve for r by dividing the second term by the first term: r = 1.65 / 1.1 = 1.5.

Now we can substitute the values of a and r into either formula to find the sum (Sn) of the first 6 terms. Let's use Formula 1:

Sn = a * (1 - r^n) / (1 - r)

In this case, n = 6 (since we want to find the sum of the first 6 terms), a = 1.1, and r = 1.5. Plugging in these values, we get:

S6 = 1.1 * (1 - 1.5^6) / (1 - 1.5)
= 1.1 * (1 - 1.5^6) / (-0.5)
= 1.1 * (-0.03125) / (-0.5)
= -0.034375 / (-0.5)
= 0.06875

Therefore, the sum of the first 6 terms of the sequence is 0.06875.

Part B: The sum of a geometric series differs from the 6th term of the sequence because it takes into account all the terms up to that point, while the 6th term only considers that specific term.

The sum of a geometric series is the result of adding up each term up to a certain point, whereas the 6th term refers only to the value of that particular term. Therefore, the sum of the series provides a cumulative total, while the 6th term represents a standalone value.

Part C: To determine the amount of protein produced in 4 hours, we need to decide whether to use the 4th term of the sequence or the sum of the first 4 terms.

If the nth term represents the total amount of protein produced in n hours, then the sum of the first n terms would represent the total amount of protein produced up to n hours. Therefore, in this case, we would use the sum of the first 4 terms to find the amount of protein produced in 4 hours.

Using either Formula 1 or Formula 2 mentioned earlier, you can substitute n = 4, a = 1.1, and r = 1.5 to find the sum of the first 4 terms.