Problem 53 - Precalculus: Mathematics for Calculus

Question

Find the sum. $$1+3+9+\cdots+2187$$

Step-by-Step Solution

Answer

The sum of the sequence is 3280.

Step 1: Identify the Type of Sequence

The sequence given is \(1, 3, 9, \ldots, 2187\). To determine the type of sequence, notice that each term is obtained by multiplying the previous term by 3. Therefore, this is a geometric sequence with the first term \(a = 1\) and a common ratio \(r = 3\).

Step 2: Determine the Number of Terms

In a geometric sequence, the \(n\)-th term is given by the formula \(a_n = a r^{n-1}\). Set the \(n\)-th term equal to 2187: \[a r^{n-1} = 2187 \]Substitute \(a = 1\) and \(r = 3\) in the equation: \[1 \times 3^{n-1} = 2187\]Solve for \(n\) by rewriting 2187 as \(3^7\):\[3^{n-1} = 3^7\]Thus, \(n-1 = 7\), so \(n = 8\). There are 8 terms in the sequence.

Step 3: Use the Formula for the Sum of a Geometric Sequence

The sum \(S_n\) of the first \(n\) terms of a geometric sequence is given by:\[S_n = a \frac{r^n - 1}{r - 1}\]Substitute \(a = 1\), \(r = 3\), and \(n = 8\):\[S_8 = 1 \frac{3^8 - 1}{3 - 1}\]Calculate \(3^8 = 6561\): \[S_8 = \frac{6561 - 1}{2}\]\[S_8 = \frac{6560}{2}\]\[S_8 = 3280\].

Step 4: Confirm the Calculation

Ensure each computation is correct. We've calculated that the number of terms is 8, and the sum formula for a geometric series has been applied correctly. The final computed sum \(S_8 = 3280\) is accurate based on the steps above.

Key Concepts

Sum of Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a fixed number called the common ratio. The sum of such a sequence is often needed in various mathematical and real-life applications. For our example above, we're given the sequence: \(1, 3, 9, \ldots, 2187\). This sequence continues by multiplying each term by 3. Determining the sum of a geometric sequence involves using a specific formula. Calculating the sum of all terms can show the growth or accumulation over the entire range of the sequence.
To find the sum of the first \(n\) terms of a geometric sequence, we apply the formula for the geometric sequence sum, which neatly simplifies the process. This formula is particularly handy because it doesn't require adding up each term individually, which can be cumbersome for long sequences.

Common Ratio

The 'common ratio' is a cornerstone in understanding geometric sequences. It's the constant factor between consecutive terms, defining the progression of the sequence. Given the sequence \(1, 3, 9, \ldots, 2187\), you spot the common ratio by dividing any term by the previous term. So, \(3 \div 1 = 3\), and further, \(9 \div 3 = 3\).
In this given sequence, the common ratio \(r\) is 3, confirming that the sequence is geometric. Recognizing the common ratio helps in identifying the pattern and predictably calculating future terms or the sequence's sum. This constant multiplier streamlines understanding the growth or decrease pattern inherent in a geometric sequence.

Number of Terms in a Sequence

Finding the number of terms in a geometric sequence is crucial for computing the sum accurately. Each term's position is pivotal because the sum formulation relies on the full term count. Using the general formula for the nth term of a geometric sequence, \(a_n = a \cdot r^{n-1}\), we can solve for \(n\) when the last term \(a_n\) is given.
Apply this to our series: With \(a = 1\), \(r = 3\), and the nth term being \(2187\), set up the equation: \(1 \cdot 3^{n-1} = 2187\). By realizing 2187 is \(3^7\), equate \(3^{n-1} = 3^7\), thereby \(n-1 = 7\) leading to \(n = 8\). Identifying these 8 terms allows us to proceed with the sum calculation confidently.

Formula for Geometric Sequence Sum

The formula for calculating the sum of a geometric sequence is a powerful tool that simplifies finding the total of all terms up to a given point. Given by \(S_n = a \frac{r^n - 1}{r - 1}\), it incorporates the first term \(a\), common ratio \(r\), and the total number of terms \(n\).
Applying this to the sequence, where \(a = 1\), \(r = 3\), and \(n = 8\) (from our previous calculations), the formula becomes \(S_8 = 1 \frac{3^8 - 1}{3 - 1}\). Calculating \(3^8 = 6561\), the expression simplifies to \(\frac{6561 - 1}{2}\), equaling \(3280\). This method elegantly compresses potentially lengthy arithmetic into a handful of steps, yielding an accurate total effortlessly.