Problem 21 - Algebra 2

Question

Write the explicit formula for each sequence. Then generate the first five terms. $$ a_{1}=10, r=-1 $$

Step-by-Step Solution

Answer

The explicit formula of the sequence is \(a_n = 10 \cdot (-1)^{(n-1)}\). The first five terms of the sequence are \(a_1 = 10\), \(a_2= -10\), \(a_3 = 10\), \(a_4 = -10\), \(a_5 = 10\).

Step 1: Write the explicit formula

To write the explicit formula for the sequence, replace \(a_1\) and \(r\) in the general formula for geometric sequences with the given values. The formula is as follows: \(a_n = a_1 \cdot r^{(n-1)}\) when \(a_1 = 10\) and \(r = -1\), the explicit formula is: \(a_n = 10 \cdot (-1)^{(n-1)}\). The minus sign to the power of \(n-1\) will interchange between positive and negative for every term, starting from positive.

Step 2: Generate the first five terms

Apply the explicit formula to find values for \(n = 1\), \(n = 2\), \(n = 3\), \(n = 4\), and \(n = 5\). These give the first five terms of the sequence.\n\nFor \(n = 1\): \(a_1 = 10 \cdot (-1)^{(1-1)} = 10 \cdot 1 = 10\)\nFor \(n = 2\): \(a_2 = 10 \cdot (-1)^{(2-1)} = 10 \cdot -1 = -10\)\nFor \(n = 3\): \(a_3 = 10 \cdot (-1)^{(3-1)} = 10 \cdot 1 = 10\)\nFor \(n = 4\): \(a_4 = 10 \cdot (-1)^{(4-1)} = 10 \cdot -1 = -10\)\nFor \(n = 5\): \(a_5 = 10 \cdot (-1)^{(5-1)} = 10 * 1 = 10\)\n\nThus the first five terms are 10, -10, 10, -10, 10.

Key Concepts

Understanding the Explicit Formula

When dealing with geometric sequences, the explicit formula is a powerful tool. It allows us to determine any term in a sequence without calculating all the preceding terms. The general explicit formula for a geometric sequence is given by:

• \(a_n = a_1 \cdot r^{(n-1)}\)

In this formula, \(a_1\) stands for the first term of the sequence, \(r\) represents the common ratio, and \(n\) is the term number we want to find.

The beauty of the explicit formula is how it condenses the process of finding terms. Simply substitute the values into the formula. For instance, if \(a_1 = 10\) and \(r = -1\), the explicit formula becomes \(a_n = 10 \cdot (-1)^{n-1}\).

This formula provides a straightforward way to figure out the value of any term in the sequence by just plugging in the term's position \(n\). No need to calculate each term one by one! "Formulas like these play a crucial role in mathematics by providing quick and efficient methods to solve problems."

Generating the First Five Terms

Now, with the explicit formula \(a_n = 10 \cdot (-1)^{n-1}\) in hand, let's generate the first five terms of the sequence.

The idea here is simple. You input different values for \(n\) to see how the sequence evolves:

• For \(n = 1\): \(a_1 = 10 \cdot (-1)^{0} = 10\)
• For \(n = 2\): \(a_2 = 10 \cdot (-1)^{1} = -10\)
• For \(n = 3\): \(a_3 = 10 \cdot (-1)^{2} = 10\)
• For \(n = 4\): \(a_4 = 10 \cdot (-1)^{3} = -10\)
• For \(n = 5\): \(a_5 = 10 \cdot (-1)^{4} = 10\)

Thus, the first five terms are 10, -10, 10, -10, and 10.

This explicit calculation showcases how alternating the sign, as dictated by the common ratio of \(-1\), affects each successive term. Using the formula gives a nice rhythm to the sequence, showing how each step matches the expected values.

The Alternating Sequence

One of the fascinating aspects of this particular sequence is its alternating nature. The sequence switches between positive and negative values, which is characteristic when the common ratio \(r\) is negative.

• What causes this alternation? It's all about the power of \((-1)\).

Depending on \(n\), this power toggles between leaving the sequence value positive when the power is even and making it negative when the power is odd.

An alternating sequence is unique in display and behavior. For this sequence, starting at \(a_1 = 10\), the terms flip signs as \(n\) increases. It is a predictable pattern: positive to negative and back to positive, every successive term.

This property adds a layer of complexity and interest, making sequences like these stand out. Sequences with alternating signs frequently appear in mathematics and other fields, serving as a perfect example of how simple rules create complex patterns. This alternating behavior is a beautiful demonstration of how geometric sequences can operate under specific conditions.