Problem 14 - College Algebra

Question

For the following exercises, write the first four terms of the sequence. $$ a_{n}=(-10)^{n}+1 $$

Step-by-Step Solution

Answer

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

Step 1: Calculate the First Term

The first term of the sequence is found by substituting \( n = 1 \) into the formula: \[ a_{1} = (-10)^{1} + 1 = -10 + 1 = -9 \] Thus, the first term is \( a_{1} = -9 \).

Step 2: Calculate the Second Term

For the second term, we substitute \( n = 2 \) into the formula: \[ a_{2} = (-10)^{2} + 1 = 100 + 1 = 101 \] Therefore, the second term is \( a_{2} = 101 \).

Step 3: Calculate the Third Term

To find the third term, substitute \( n = 3 \) into the sequence formula: \[ a_{3} = (-10)^{3} + 1 = -1000 + 1 = -999 \] So, the third term is \( a_{3} = -999 \).

Step 4: Calculate the Fourth Term

For the fourth term, substitute \( n = 4 \) into the sequence formula: \[ a_{4} = (-10)^{4} + 1 = 10000 + 1 = 10001 \] Thus, the fourth term is \( a_{4} = 10001 \).

Key Concepts

Arithmetic Sequences

Arithmetic sequences are like adding the same number over and over again to get to the next number in the sequence. Think of them as having a steady pattern of addition. In algebra, you often see them written using a formula. This formula helps you find any term in the sequence.

• Each term after the first is produced by adding a constant called the "common difference" to the previous term.
• If the common difference is positive, each term is bigger than the one before it.
• If the common difference is negative, each term is smaller than the one before it.

Unlike the sequence in the exercise above, which does not have a constant difference between terms, understanding arithmetic sequences is important. They are simple, predictable, and form the basis for more complex patterns. Identifying them helps with understanding sequences better, even when the sequence follows more complicated rules, like those involving powers.

Exponents

Exponents are a way to show how many times a number, called the base, is multiplied by itself. In mathematical notation, you’ll see them expressed as a small, raised number.For example, in the sequence \[ a_{n} = (-10)^{n} + 1 \]the -10 is the base, and \( n \) is the exponent. Here’s how exponents work:

• A positive exponent means you multiply the base by itself that many times. For example, \(-10^2 = (-10) \times (-10) = 100\).
• A negative base raised to an odd power stays negative, such as \((-10)^3 = (-10) \times (-10) \times (-10) = -1000\).
• A negative base raised to an even power becomes positive, because multiplying two negative numbers results in a positive number.

Understanding exponents is crucial in algebra and beyond. They compactly represent large calculations and appear in various scientific and financial formulas. In sequences, exponents can create dramatic changes between terms, making patterns more dynamic.

Recursive Formulas

Recursive formulas are a way to define sequences where each term is based on the previous terms. This usually creates a chain reaction, where one term builds upon another. It’s like a recipe where each step depends on the last one being completed.

• A recursive formula has two parts: a starting value (often called the "base case") and a rule for finding each subsequent term from the one or few preceding it.
• For example, in post-calculation, once you know \( a_1 \), you could derive \( a_2 \) and onwards if this sequence was recursive (but the given sequence has an explicit formula).
• Recursive sequences need careful calculation as missing one term can throw off ones that follow, thus usually require checking using the formula.

While the sequence from the exercise is not given in a recursive form, understanding these can help in breaking down complex sequences into simpler components. They show the interconnected nature of each term in a sequence and highlight dependency between sequential elements.