Problem 29 - Algebra and Trigonometry

Question

Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 14 terms of the geometric sequence: \(-\frac{3}{2}, 3,-6,12, \dots\)

Step-by-Step Solution

Answer

The sum of the first 14 terms of the given geometric sequence is -16383.

Step 1: Identify the first term and the common ratio

The first term \(a\) of our geometric sequence is \(-3/2\). The common ratio \(r\) is found by dividing the second term by the first term, or the third term by the second term, etc. Hence, \(r = 3/(-3/2) = -2.

Step 2: Apply the formula for the sum of the first n terms

Now that we have our \(a\), \(r\), and \(n\) (which is 14), we can apply the formula for the sum of the first n terms of a geometric sequence: \(S_n = a(1 - r^n)/(1 - r)\). So, \(S_{14} = -3/2 * (1 - (-2)^{14})/(1 - (-2))\).

Step 3: Calculate the sum

Calculate the given expression to find \(S_{14}\). This simplifies to \(S_{14}=-3/2*(1 - 16384)/3=-3/2*-16383/3\). Calculate this value to find the sum.

Key Concepts

Sum of Sequence

The sum of a sequence in the context of geometric sequences refers to the total of the terms up to a certain point. A geometric sequence is a set of numbers where each term is derived by multiplying the previous term by a constant, known as the common ratio.
To find the sum of the sequence, especially when it involves a large number of terms, we use the geometric series formula. This formula helps to simplify calculations and provides the exact sum of all terms up to the specified number.
The sum formula for the first \(n\) terms of a geometric sequence is:

• \( S_n = a \frac{(1 - r^n)}{1 - r} \)

where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms to be summed. This formula ensures that even if the sequence progresses to very large or very small numbers, we can efficiently find their sum.

First Term

In any geometric sequence, the first term is crucial as it lays the foundation for all subsequent calculations. It is represented by \(a\). This value sets the initial point from which the series of calculations begins.
In our example exercise, the first term \(a\) is \(-\frac{3}{2}\). Knowing the first term allows us to understand the starting point of the sequence, and it is used directly in the formula for calculating the sum of the sequence.
Think of the first term like the anchor of a chain; without it, the links (or other terms in the sequence) have no starting point.

Common Ratio

The common ratio is a fundamental element in a geometric sequence. Represented by \(r\), it defines the relationship between consecutive terms. This ratio remains constant throughout the sequence.
To determine the common ratio, we divide any term in the sequence by its preceding one. For instance, the ratio in our example is found by dividing the second term, \(3\), by the first term, \(-\frac{3}{2}\), resulting in \(-2\). This tells us how each term grows or shrinks compared to the last one.
A constant common ratio allows us to predict the sequence's behavior and also plays a critical role in applying the geometric series formula for the sum.

Geometric Series Formula

The geometric series formula is a powerful tool for finding the sum of the terms in a geometric sequence. It is particularly useful when dealing with many terms, as it simplifies potentially cumbersome calculations.

• The formula is: \( S_n = a \frac{(1 - r^n)}{1 - r} \)

In this formula:

• \( S_n \) is the sum of the first \(n\) terms.
• \( a \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the number of terms to be summed.

The geometric series formula elegantly reduces the complexities of manual summation, providing a direct computation path to the sum. It's one of the key benefits of recognizing a sequence as geometric, as it distills a potentially complex problem into a simple arithmetic operation.