Problem 45 - Algebra and Trigonometry

Question

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$ a_{3}=28, \quad a_{6}=224, \quad n=6 $$

Step-by-Step Solution

Answer

The partial sum of the first 6 terms is 441.

Step 1: Understanding the Geometric Sequence Formula

A geometric sequence has the general term given by \(a_n = a_1 \, r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio. We need to find these to solve for the partial sum \(S_n\) of the first 6 terms.

Step 2: Find the Common Ratio (r)

We know \(a_3 = a_1 \, r^2 = 28\) and \(a_6 = a_1 \, r^5 = 224\). Divide \(a_6\) by \(a_3\) to eliminate \(a_1\):\[\frac{a_6}{a_3} = \frac{a_1 \, r^5}{a_1 \, r^2} = r^3 = \frac{224}{28}\]\[ r^3 = 8 \]Taking the cube root gives \(r = 2\).

Step 3: Find the First Term (a1)

Substitute \(r = 2\) into \(a_3 = a_1 \, 2^2 = 28\): \[a_1 \, 4 = 28\]\[a_1 = 7\].

Step 4: Calculate the Partial Sum (Sn) for n=6

The partial sum \(S_n\) of a geometric series is given by:\[S_n = a_1 \frac{r^n - 1}{r - 1}\]Substitute \(a_1 = 7\), \(r = 2\), and \(n = 6\):\[S_6 = 7 \frac{2^6 - 1}{2 - 1} = 7 \times (64 - 1)\]\[S_6 = 7 \times 63 = 441\].

Step 5: Verify the Solution

Check that calculations are consistent with the given terms to ensure no errors.\( a_3 = a_1 \, r^2 = 7 \, \times \, 4 = 28\) and \( a_6 = a_1 \, r^5 = 7 \, \times \, 32 = 224\) are correct.

Key Concepts

Partial Sum

In a geometric sequence, the partial sum refers to the sum of a specific number of terms starting from the first term. It can be thought of as the cumulative sum of terms up to a certain point in the sequence.
When calculating the partial sum, we use a formula that simplifies the process, especially for sequences with large numbers of terms. To find the partial sum for a geometric sequence with a known number of terms, we apply the partial sum formula of a geometric series:

• Given by: \[ S_n = a_1 \frac{r^n - 1}{r - 1} \] where \( S_n \) is the partial sum of the first \( n \) terms, \( a_1 \) is the first term, and \( r \) is the common ratio.
• For our exercise, since \( n = 6 \), partially sum the first six terms to get \( S_6 \).
• Once the first term and the common ratio are known, inserting them into the formula gives the partial sum directly.

Understanding this formula is key to mastering arithmetic involving geometric sequences.

Common Ratio

The common ratio is a critical component in a geometric sequence. It's the factor by which each term is multiplied to get the subsequent term in the sequence. Determining the common ratio is essential as it sets the pattern for the sequence.
To find the common ratio, you can use any two consecutive terms in the sequence and divide the term by its predecessor. Alternatively, as shown in the solution, non-consecutive terms can be used, leveraging the equation:

• From terms \(a_3\) and \(a_6\) in our exercise, we use \( \frac{a_6}{a_3} = r^3 \).
• This method gives \( r^3 = 8 \), hence \( r = 2 \), when taking the cube root.

Finding this ratio helps in establishing the relationships between terms, which is crucial for calculating sums and other operations.

First Term Calculation

The first term of a geometric sequence is denoted as \( a_1 \) and is key to determining other properties of the sequence. Once the common ratio \( r \) is known, the first term can be calculated using any known term in the sequence.
In practice, like in the exercise:

• Use the equation of a later term \( a_n = a_1 r^{n-1} \).
• Plug in the known values from the sequence: \( a_3 = a_1 \cdot r^2 = 28 \).
• Substitute \( r = 2 \), yielding: \( a_1 \cdot 4 = 28 \) or \( a_1 = 7 \).

This calculation allows us to pin down the starting point of the sequence, which is necessary when calculating the sum of its terms.

Geometric Series Formula

The geometric series formula is a powerful tool that allows us to sum terms of a geometric sequence swiftly and accurately. This formula is particularly useful when dealing with series that have a significant number of terms.
For a geometric series, the sum of the first \( n \) terms is calculated using:

• The formula: \[ S_n = a_1 \frac{r^n - 1}{r - 1} \].
• The terms \( a_1 \) and \( r \) must be known to evaluate \( S_n \).
• In our exercise, with \( a_1 = 7 \), \( r = 2 \), and \( n = 6 \), we got \( S_6 = 7 \times 63 = 441 \).

This formula makes it straightforward to find the sum without needing to manually add each term, and is crucial for solving similar problems efficiently. Understanding its application helps greatly with broader mathematical concepts involving sequences and series.