Problem 14 - College Algebra

Question

Write the first four terms of the sequence. $$a_{n}=(-10)^{n}+1$$

Step-by-Step Solution

Answer

The first four terms of the sequence are -9, 101, -999, and 10001.

Step 1: Understand the Formula

The formula to find the terms of the sequence is given by \( a_n = (-10)^n + 1 \). This means each term in the sequence is calculated by raising \(-10\) to the power \(n\) and then adding 1.

Step 2: Calculate the First Term

Substitute \( n = 1 \) into the formula: \[ a_1 = (-10)^1 + 1 = -10 + 1 = -9 \]So, the first term \( a_1 \) is -9.

Step 3: Calculate the Second Term

Substitute \( n = 2 \) into the formula: \[ a_2 = (-10)^2 + 1 = 100 + 1 = 101 \]Therefore, the second term \( a_2 \) is 101.

Step 4: Calculate the Third Term

Substitute \( n = 3 \) into the formula: \[ a_3 = (-10)^3 + 1 = -1000 + 1 = -999 \]This means, the third term \( a_3 \) is -999.

Step 5: Calculate the Fourth Term

Substitute \( n = 4 \) into the formula: \[ a_4 = (-10)^4 + 1 = 10000 + 1 = 10001 \]Thus, the fourth term \( a_4 \) is 10001.

Key Concepts

Calculating Terms of a Sequence

In the realm of mathematics, sequences are essentially ordered lists of numbers. Each number in the sequence is referred to as a "term." To find terms of a given sequence, you often use a formula specific to that sequence. A formula provides a rule for calculating each term based on its position in the sequence, commonly denoted by a variable such as \( n \). In our sequence, the formula is given by \( a_n = (-10)^n + 1 \). This formula tells us that each term is determined by evaluating \((-10)^n\) and adding 1.
To generate the terms in order, we'll substitute \( n \) with its respective term number (1, 2, 3, etc.). This systematic substitution helps us understand how the sequence progresses and what the actual terms are. Always remember that getting comfortable with substituting different values into the expression is a core skill when dealing with sequences.

Arithmetic Operations

Arithmetic operations are basic mathematical functions we perform on numbers, such as addition, subtraction, multiplication, and division. When calculating terms in sequences, these operations are essential. In our sequence formula \( a_n = (-10)^n + 1 \), mostly two of these operations come into play:

• Exponentiation: Raising a number to a power, which is the core operation defining our sequence.


• Addition: After exponentiating \(-10\), we add 1 to obtain each term in the sequence.

Understanding these basic operations allows us to precisely follow the instructions of a function. Always perform operations as dictated by the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). This order ensures that calculations yield the correct result.

Negative and Positive Numbers

When we deal with sequences, we'll often encounter both negative and positive numbers, as seen in our sequence formula \( a_n = (-10)^n + 1 \). Negative numbers reflect quantities less than zero, while positive numbers range above zero. How they interact depends largely on arithmetic rules:

• Multiplying two negative numbers results in a positive number.


• Multiplying a positive number by a negative number results in a negative number.


• Adding two numbers follows straightforward rules: two positives or two negatives add normally, while one positive and one negative involve subtracting the smaller from the bigger and keeping the sign of the larger absolute value.

These rules underscore the peculiarity of our sequence. When \( n \) is odd, \((-10)^n\) evaluates to a negative number, and when it's even, it evaluates to a positive number. This peculiarity leads to alternating signs in the sequence terms, producing an intriguing pattern.

Exponents in Sequences

Exponents are mathematical notations indicating the number of times a number, known as the "base," is multiplied by itself. In our sequence formula, \( (-10)^n \), the base is \(-10\) and the exponent is \( n \), which denotes the term number. Here’s how exponents work in this context:

• If \( n \) is 1, the expression is \((-10)^1 = -10\).


• If \( n \) is 2, the expression is \((-10)^2 = 100\). Since it's squared, the negative sign disappears.


• Continuing this pattern, if \( n \) is odd, the result is negative, and if \( n \) is even, it is positive.

The exponentiation process controls the size and sign of the sequence terms significantly, introducing alternating positivity and negativity based on whether the term number is odd or even. This oscillating pattern is common in sequences using exponents and essential in understanding how such sequences behave mathematically.