Problem 8 - College Algebra

Question

For the following exercises, write the first four terms of the sequence. $$ a_{n}=-(-5)^{n-1} $$

Step-by-Step Solution

Answer

The first four terms of the sequence are -1, 5, -25, 125.

Step 1: Determine the first term

To find the first term of the sequence, substitute \( n = 1 \) into the formula: \[a_1 = -(-5)^{1-1} = -(-5)^0\]Since any number raised to the power of zero is 1,\(-(-5)^0 = -1\). Therefore, the first term \( a_1 \) is \(-1\).

Step 2: Calculate the second term

Now substitute \( n = 2 \) into the formula: \[a_2 = -(-5)^{2-1} = -(-5)^1\]This simplifies to \( -(-5) = 5 \). Therefore, the second term \( a_2 \) is \( 5 \).

Step 3: Find the third term

Substitute \( n = 3 \) into the formula: \[a_3 = -(-5)^{3-1} = -(-5)^2\]\((-5)^2 = 25\), so \(-25 = -25\). Therefore, the third term \( a_3 \) is \(-25\).

Step 4: Determine the fourth term

Substitute \( n = 4 \) into the formula: \[a_4 = -(-5)^{4-1} = -(-5)^3\]\((-5)^3 = -125\), so \(-(-125) = 125\). Therefore, the fourth term \( a_4 \) is \( 125 \).

Key Concepts

Sequence Terms

In sequences, each number is a term. Identifying these terms is crucial once you have the sequence formula. A formula like \(a_n = -(-5)^{n-1}\) helps you find each sequence term based on its position \(n\) in the sequence. By substituting different values of \(n\), we can calculate the sequence terms systematically.

To get the first four terms of the sequence:

• Substitute \(n = 1\) to find the first term.
• Substitute \(n = 2\) to find the second term.
• Continue substituting \(n = 3\) and \(n = 4\) for the third and fourth terms.

This method provides clarity and consistency in determining sequence terms, making the process of analyzing sequences much more manageable.

Exponential Functions

Exponential functions are mathematical expressions where variables appear as exponents. They are a powerful tool in sequence analysis and other mathematical fields. In the sequence formula \(a_n = -(-5)^{n-1}\), \(-5\) is the base and it is raised to \((n-1)\) power.

Understanding how exponential functions behave when the base is negative can be insightful:

• When the exponent is even, as in \((-5)^2\), the result is positive.
• When the exponent is odd, as in \((-5)^3\), the result is negative.

This distinction is useful as it helps predict the signs of the terms in the sequence, showing an alternating pattern of positive and negative values.

Power of Integers

Exponents describe how many times to use the number in a multiplication. When integers are raised to a power, they grow or shrink rapidly. The base \(-5\) in the power of integers within our sequence formula alters the size dramatically based on the exponent value.

Here's how it works:

• As you increase the exponent, the absolute value of the result becomes larger if the base is greater than 1.
• For instance, \((-5)^0\) equals 1, \((-5)^1\) equals -5, and increasing to \((-5)^2\) results in 25.

Notice the repeated multiplication significantly changes the value, especially noticeable as the sequence progresses. Recognizing this helps you anticipate the magnitude of sequence terms in such exponential contexts.