Finding The Explicit Formula For A Recursive Sequence

Hey math enthusiasts! Today, we're diving into a cool problem that combines recursive and explicit formulas. We'll be figuring out which explicit formula matches up with a given recursive one. This stuff is super important for understanding sequences and patterns, so let's get started.

Understanding the Problem

The core of our problem is understanding the relationship between a recursive formula and its corresponding explicit formula. A recursive formula tells us how to find a term in a sequence based on the previous term(s). It's like a set of instructions that build upon each other. Think of it as a chain reaction. An explicit formula, on the other hand, gives us a direct way to find any term in the sequence without having to calculate all the terms before it. It's like a shortcut that takes us straight to the answer. The given information is: a_1 = 1, a_n = -6a_{n-1}.

Analyzing the Given Recursive Formula

First off, let's break down the recursive formula we've got: a_1 = 1, a_n = -6a_{n-1}. This tells us a couple of things:

• a_1 = 1: The first term in our sequence is 1. This is our starting point.
• a_n = -6a_{n-1}: To find any term a_n, we multiply the previous term a_{n-1} by -6. This is the heart of our recursive definition. It shows us how each term depends on the one before it.

Now, let's see how the sequence unfolds:

• a_1 = 1 (given)
• a_2 = -6 * a_1 = -6 * 1 = -6
• a_3 = -6 * a_2 = -6 * -6 = 36
• a_4 = -6 * a_3 = -6 * 36 = -216

Notice the pattern? Each term is multiplied by -6 to get the next term. This type of sequence is called a geometric sequence. Geometric sequences are characterized by a constant ratio between consecutive terms. In our case, the common ratio is -6.

Evaluating the Answer Choices

Alright, let's examine the answer choices to see which one fits our sequence:

A. a_n = 1(-6)^{n-1}

• Let's check the first few terms:
  • a_1 = 1(-6)^{1-1} = 1(-6)^0 = 1 * 1 = 1 (Correct)
  • a_2 = 1(-6)^{2-1} = 1(-6)^1 = 1 * -6 = -6 (Correct)
  • a_3 = 1(-6)^{3-1} = 1(-6)^2 = 1 * 36 = 36 (Correct)

B. a_n = -6 + (n-1)

• Let's check the first few terms:
  • a_1 = -6 + (1-1) = -6 + 0 = -6 (Incorrect)

C. a_n = 1 + (n-1)^6

• Let's check the first few terms:
  • a_1 = 1 + (1-1)^6 = 1 + 0 = 1 (Correct)
  • a_2 = 1 + (2-1)^6 = 1 + 1 = 2 (Incorrect)

D. a_n = -6(1)^{n-1}

• Let's check the first few terms:
  • a_1 = -6(1)^{1-1} = -6(1)^0 = -6 * 1 = -6 (Incorrect)

The Correct Answer

Based on our analysis, the correct answer is A. a_n = 1(-6)^{n-1}. This formula accurately represents the given recursive sequence, where each term is obtained by multiplying the previous term by -6, and the first term is 1. The formula gives the correct values for a_1, a_2, and a_3, matching the pattern we observed in our calculations. This formula represents a geometric sequence with a first term of 1 and a common ratio of -6.

Deeper Dive into Geometric Sequences

Let's get a little deeper on geometric sequences. As we've seen, geometric sequences are sequences where each term is found by multiplying the previous term by a constant value. This constant value is known as the common ratio, often denoted by 'r'. The general form of an explicit formula for a geometric sequence is: a_n = a_1 * r^(n-1), where:

• a_n is the nth term
• a_1 is the first term
• r is the common ratio
• n is the term number

In our problem, a_1 = 1 and r = -6. Substituting these values into the general formula gives us: a_n = 1 * (-6)^(n-1), which simplifies to a_n = (-6)^(n-1). This is equivalent to the correct answer choice A. This formula is a powerful tool because it allows us to find any term in the sequence directly, without having to calculate all the preceding terms. It's a fundamental concept in algebra and is used in a wide range of applications, from finance to computer science.

Why Other Choices are Incorrect

Let's clarify why the other answer choices don't work:

• B. a_n = -6 + (n-1): This represents an arithmetic sequence, where a constant value is added to each term. This doesn't match the geometric pattern of our given sequence.
• C. a_n = 1 + (n-1)^6: This is neither arithmetic nor geometric. The terms don't follow a constant ratio or a constant difference, so it doesn't align with our recursive definition.
• D. a_n = -6(1)^{n-1}: This simplifies to a_n = -6, meaning every term in the sequence would be -6. This doesn't match our initial term of 1 or the subsequent terms in the sequence.

The Power of Explicit Formulas

Explicit formulas are super handy because they give us a direct way to calculate any term in a sequence. Imagine you need to find the 100th term of a sequence. Using the recursive formula would mean calculating all 99 terms before it! But with the explicit formula, you just plug in n=100 and bam, you have your answer. This highlights the efficiency and usefulness of explicit formulas.

Final Thoughts

So there you have it, folks! We've successfully matched a recursive formula with its explicit counterpart. Remember, understanding the difference between recursive and explicit formulas is key to mastering sequences. Keep practicing, and you'll become a pro at these problems in no time. If you enjoyed this explanation, give it a thumbs up and stay tuned for more math adventures! Don’t forget to subscribe to our channel for more awesome math content. See ya next time!