Problem 3 - Algebra 2

Question

Evaluate the finite series for the specified number of terms. $$ 3+6+12+\ldots ; n=7 $$

Step-by-Step Solution

Answer

The sum of the first 7 terms of the geometric series is 765.

Step 1: Identify the First Term and Common Ratio

Looking at the given series, we see that the first term (a) is 3. The common ratio (r) can be found by dividing the second term by the first term or the third term by the second term, etc. Doing this yields \( r = \frac{6}{3} = 2\).

Step 2: Apply the Formula for Sum of Geometric Series

Apply the formula \( S_n = \frac{a(r^n - 1)}{r - 1}\) to calculate the sum of first 7 terms. Substituting the value of 'a', 'r' and 'n' into the formula, \( S_n = \frac{3* (2^7 - 1)}{2 - 1}\).

Step 3: Calculate the Result

Perform the calculations for \(2^7\), subtraction, multiplication and division in the formula obtained in step 2. This will give the sum of the first 7 terms of the geometric series.

Key Concepts

Finite Series

A finite series is a collection of numbers added together, where the sequence has a specific number of terms. This is in contrast to an infinite series, which continues indefinitely.

In our example, the sequence of 3, 6, 12, and so on forms a finite series because it is evaluated up to 7 terms, as specified by the value of \(n\).

Finite series play an important role in mathematics because they allow us to deal with calculations involving a limited number of elements. The concept requires you to know the number of terms in advance to perform any calculations, which simplifies the process of finding sums or other operations.

Common Ratio

The common ratio is a vital part of understanding geometric series. It is the factor by which each term of the series is multiplied to get the next term.

For a geometric sequence, like the one in the exercise, the sequence of terms
3, 6, 12, 24, ... has a common ratio of 2.

• It is calculated by dividing one term by its preceding term.
• For example, \( r \) is \( \frac{6}{3} = 2 \).

Knowing the common ratio helps in determining entire series and is crucial in calculating the sum using formulas designed for geometric series.

Sum Formula

To find the sum of a finite geometric series, we use the sum formula:\[S_n = \frac{a(r^n - 1)}{r - 1}\]Here:

• \( S_n \) is the sum of the series up to \( n \) terms.
• \( a \) is the first term of the series.
• \( r \) is the common ratio.
• \( n \) is the number of terms in the series.

In our example, to find the sum of the first 7 terms, plug \( a = 3 \), \( r = 2 \), and \( n = 7 \) into the formula:\[S_7 = \frac{3 (2^7 - 1)}{1} = 3(127) = 381\]This formula simplifies the process and allows for quick calculation without needing to add each term individually.

Term Evaluation

Evaluating individual terms of a geometric sequence can also be straightforward, provided you know the first term and the common ratio.Each term in a geometric series can be determined using:\[a_n = a \cdot r^{(n-1)}\]

• \( a_n \) is the \( n^{th} \) term.
• \( a \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the term number.

For example, to find the 3rd term when \( a = 3 \) and \( r = 2 \):
\[a_3 = 3 \cdot 2^{(3-1)} = 3 \cdot 4 = 12\]Using this formula can quickly determine any specific term in the series. This is crucial for checking calculations or finding missing terms when working with finite series.