Problem 15 - Algebra 2

Question

Find \(S_{n}\) for each geometric series described. $$ a_{1}=-8, a_{6}=-256, r=2 $$

Step-by-Step Solution

Answer

The sum of the first 6 terms, \( S_6 \), is -504.

Step 1: Identify Given Values

We are given the first term of the geometric series, \( a_1 = -8 \), the common ratio \( r = 2 \), and the sixth term \( a_6 = -256 \).

Step 2: Find the General Term Formula

The formula for the general term \( a_n \) of a geometric series is \( a_n = a_1 \, r^{n-1} \). Substitute the given values: \( a_6 = -8 \, \times \, 2^{6-1} \).

Step 3: Solve for \( a_6 \)

Calculate the sixth term to verify given data: \( a_6 = -8 \, \times \, 2^5 = -8 \, \times \, 32 = -256 \). The calculation is consistent with the given \( a_6 \).

Step 4: Apply Sum Formula for Geometric Series

The sum \( S_n \) of the first \( n \) terms of a geometric series is given by \( S_n = a_1 \frac{r^n - 1}{r - 1} \).

Step 5: Substitute Known Values and Solve

Substitute \( a_1 = -8 \), \( r = 2 \), and \( n = 6 \):\[ S_6 = -8 \frac{2^6 - 1}{2 - 1} \].

Step 6: Calculate Powers and Simplify

Calculate \( 2^6 = 64 \) and substitute: \[ S_6 = -8 \frac{64 - 1}{1} = -8 \times 63 \].

Step 7: Compute the Final Sum

Finally, compute \( S_6 \): \( -8 \times 63 = -504 \).

Key Concepts

First Term of Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant known as the common ratio. The journey starts with the first term, often denoted as \( a_1 \). This first term sets the stage for the entire sequence. In our exercise, the first term \( a_1 \) is clearly identified as \(-8\).

Think of \( a_1 \) as the launch pad of a rocket. It provides the initial "push" for the sequence and is vital for determining subsequent terms.

If you know the first term, you can begin uncovering other properties or terms of the series using formulas. In any sequence, especially a geometric one, having a firm grip on \( a_1 \) gives you a solid starting point.

Common Ratio

The common ratio of a geometric sequence is fundamental because it determines how the sequence behaves. This ratio, denoted as \( r \), is the factor by which we multiply each term to get to the next term. In our problem, this common ratio is given as \( 2 \).

The common ratio:

• If \( r > 1 \), each term increases in value.
• If \( 0 < r < 1 \), each term decreases, getting closer to zero.
• If \( r < 0 \), the terms alternate in sign.

In our exercise, since \( r = 2 \) and is greater than one, the sequence will grow exponentially. Understanding \( r \) helps you predict how a sequence will expand or contract.

Sum of Geometric Series

Calculating the sum of a geometric series is a crucial skill. It helps evaluate the total of all terms up to a particular point. The formula to find this sum \( S_n \) is \( S_n = a_1 \frac{r^n - 1}{r - 1} \). This formula is exceptionally useful for financial calculations and predicting future outcomes based on a pattern.

In the exercise, we are asked to find \( S_6 \):

Substitute the known values into the sum formula:

• First term \( a_1 = -8 \)
• Common ratio \( r = 2 \)
• Number of terms \( n = 6 \)

So, the formula becomes \(-8 \times \frac{2^6 - 1}{2 - 1} = -8 \times 63\), resulting in a sum of \(-504\). This tells us the sum of the first six terms of the sequence.

General Term Formula

Finding the general term of a geometric sequence is like having a recipe that allows you to make any term in the sequence. The formula is \( a_n = a_1 \, r^{n-1} \). Knowing this allows you to compute any specific term without needing to list all previous terms.

In our specific exercise, to reaffirm the sequence, verify that the sixth term \( a_6 \) is calculated by substituting the known values into the general term formula: \( a_6 = -8 \, \times \, 2^{6-1} \). This amplication of the common ratio shows: \( a_6 = -8 \, \times \, 32 = -256 \). Coordinates of each term can be efficiently calculated with this formula. It simplifies the process as you transcend through the sequence, understanding each term's value and role.