Problem 49 - Precalculus: Mathematics for Calculus

Question

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$a=5, r=2, n=6$$

Step-by-Step Solution

Answer

The partial sum \( S_6 \) is 315.

Step 1: Understand the Formula for a Geometric Series

The partial sum of a geometric series can be found using the formula: \( S_n = a \frac{r^n - 1}{r - 1} \), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

Step 2: Substitute the Given Values into the Formula

With \(a = 5\), \(r = 2\), and \(n = 6\), substitute these into the formula: \[ S_6 = 5 \frac{2^6 - 1}{2 - 1} \].

Step 3: Calculate the Power of the Common Ratio

Calculate \(2^6\), which is 64. This provides \[ S_6 = 5 \frac{64 - 1}{1} \].

Step 4: Simplify the Expression

Subtract 1 from 64 to get 63: \[ S_6 = 5 \times 63 \].

Step 5: Compute the Product

Calculate the product \(5 \times 63 = 315\). Thus, the partial sum \(S_6 = 315\).

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers arranged in a specific order where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. This core concept is fundamental to understanding sequences. Besides the first term, the common ratio is the defining feature of a geometric sequence.
For example, in the exercise, we have a first term of 5. Using the common ratio of 2, each term is twice the preceding term. Hence, the sequence begins as 5, 10, 20, and so on. To grasp how this pattern comes to be, think of it like a chain of repeated multiplications. It is this repetitive pattern that distinguishes geometric sequences from others.

Partial Sum

The partial sum of a geometric sequence is the total of a specified number of consecutive terms from this sequence. If you have a sequence and you want to know the sum of the first few elements, you calculate the partial sum.
A specific formula is used to find this sum, and it is grounded in the following logic: You add up the terms in the sequence to get a subtotal — but only up to the term you're interested in. For the sequence in the exercise, to find the sum of the first 6 terms, we use the formula for the partial sum:

• The formula is: \( S_n = a \frac{r^n - 1}{r - 1} \).
• By substituting for \(a = 5\), \(r = 2\), and \(n = 6\), the calculation gives us 315.

Using this formula allows for an efficient computation of the sum without needing to add each number individually.

Common Ratio

The common ratio in a geometric sequence is a constant number that each term is multiplied by to get to the next term. It plays the role of a multiplier which shapes the sequence’s progression.
In our exercise, the common ratio is 2, which implies every term is '2' times its previous one. This uniform multiplier is crucial as it maintains the consistency required for a sequence to be geometric. Understanding the common ratio also helps in predicting future terms and in calculating the sum of terms, as we did in finding the partial sum.

First Term

The first term is the starting point of a geometric sequence. It is symbolized as \(a\) in the formula for both individual terms and sums.
In the exercise at hand, the first term is 5, inferring that from this point each subsequent number is derived by continuous multiplication by the common ratio. This term is essential because it determines the baseline of the sequence. The size of the first term can drastically affect the sequence's growth and the resultant sums, which is noticeable in our result of the partial sum being calculated.