Problem 43 - Algebra and Trigonometry

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 Sequence Type

Notice that the sequence given is: \(1, 3, 9, \cdots, 2187\). This sequence is a geometric sequence because each term is obtained by multiplying the previous term by a common ratio. To find the common ratio \(r\), divide the second term by the first term, \(r = \frac{3}{1} = 3\).

Step 2: Determine Number of Terms

To find the number of terms \(n\), use the formula for the \(n\)-th term of a geometric sequence: \(a_n = a_1 \cdot r^{n-1}\). The last term given is 2187, so we have \(2187 = 1 \cdot 3^{n-1}\). Solving for \(n\), we get \(3^{n-1} = 2187\). Recognizing 2187 as a power of 3, we find that 2187 equals \(3^7\). Thus, \(n = 8\) because \(n-1 = 7\).

Step 3: Use the Sum Formula for Geometric Series

The sum \(S_n\) of the first \(n\) terms of a geometric sequence is given by the formula \(S_n = a_1 \cdot \frac{r^n - 1}{r - 1}\), where \(a_1 = 1\), \(r = 3\), and \(n = 8\). Substitute these values into the formula: \[ S_8 = 1 \cdot \frac{3^8 - 1}{3 - 1} \].

Step 4: Compute the Sum

First calculate \(3^8\): \(3^8 = 6561\). Then plug this result into the sum formula: \[ S_8 = \frac{6561 - 1}{2} = \frac{6560}{2} = 3280 \].

Step 5: Conclusion

The sum of the sequence \(1 + 3 + 9 + \cdots + 2187\) is \(3280\).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial element. It determines how each term progresses from the previous one. To find the common ratio, take any term in the sequence and divide it by the preceding term. For example, in the sequence \(1, 3, 9, \ldots, 2187\), the common ratio \(r\) can be found as follows:
\[ r = \frac{3}{1} = 3 \]
This means that each term is three times the term before it. The common ratio can be any non-zero number, and it is constant throughout the sequence.
Understanding the common ratio is vital, as it makes it possible to find other elements, such as future terms or the series' sum.

Sum of a Geometric Series

The sum of a geometric series refers to the total of all terms in a sequence. To find this sum, we use a formula that incorporates the first term, the common ratio, and the number of terms.
The formula to determine the sum of the first \(n\) terms \(S_n\) is:
\[ S_n = a_1 \cdot \frac{r^n - 1}{r - 1} \]
where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

• For our sequence, \(a_1 = 1\), \(r = 3\), and \(n = 8\).
• Thus, the sum is calculated as:

\[ S_8 = 1 \cdot \frac{3^8 - 1}{3 - 1} = \frac{6561 - 1}{2} = 3280 \]
This approach allows you to quickly find the total of any geometric sequence without manually adding each term.

Number of Terms in a Sequence

Knowing the number of terms in a geometric sequence involves a straightforward method. This is essential when employing the sum formula, as it requires the number of terms. To locate the number of terms \(n\), use the formula for the \(n\)-th term:
\[ a_n = a_1 \cdot r^{n-1} \]
With \(a_n = 2187\), the final term, \(a_1 = 1\), and \(r = 3\), we set up the equation:
\[ 2187 = 1 \cdot 3^{n-1} \]
We need to understand that 2187 equals \(3^7\), indicating \(n-1 = 7\), hence \(n = 8\).
Recognizing the number of terms helps both in calculating the sum and in understanding the sequence's scope. Without knowing \(n\), more complex operations on the sequence become challenging.