Finding The 3rd Term Of A Sequence With A Given Rule

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Embarking on a mathematical journey often involves deciphering patterns and sequences. In this exploration, we tackle a fascinating problem that requires us to reverse-engineer a sequence, utilizing the given term-to-term rule to unveil the elusive 3rd term. Our initial clue is that the 5th term of the sequence is 489, and the sequence follows a unique rule: subtract 2, then multiply by 3. This intricate dance of operations makes the task both challenging and rewarding. To conquer this problem, we'll employ a systematic approach, carefully reversing the operations to navigate back to the 3rd term. Join us as we unravel the mysteries of this sequence and discover the solution together.

Understanding the Sequence and the Challenge

Before diving into the calculations, it's crucial to grasp the essence of sequences and term-to-term rules. A sequence, in mathematical terms, is an ordered list of numbers or other mathematical objects. Each element in the sequence is called a term, and the term-to-term rule dictates how we transition from one term to the next. In our specific case, the term-to-term rule is a two-step process: subtract 2, then multiply by 3. This means that to get the next term in the sequence, we first subtract 2 from the current term, and then multiply the result by 3.

The challenge lies in the fact that we are given the 5th term and the term-to-term rule, but we need to find the 3rd term. This requires us to work backward, undoing the operations to retrace the sequence. This process is akin to solving a puzzle, where we need to carefully analyze the clues and reverse the steps to reach the desired solution. The complexity arises from the order of operations – we need to reverse the multiplication before undoing the subtraction. This meticulous approach will ensure that we accurately find the 3rd term of the sequence.

The Strategy: Reversing the Operations

To find the 3rd term, we'll systematically reverse the operations defined in the term-to-term rule. Since the rule involves subtracting 2 and then multiplying by 3, we need to undo these operations in reverse order. This means we'll first reverse the multiplication by 3, and then reverse the subtraction of 2. This approach ensures that we accurately backtrack through the sequence, step by step.

Starting from the 5th term, which is 489, we'll first reverse the multiplication by 3. To do this, we divide 489 by 3. This calculation will give us the value before the multiplication occurred. Next, we'll reverse the subtraction of 2 by adding 2 to the result of the division. This will give us the 4th term in the sequence. We'll then repeat this process – dividing by 3 and adding 2 – to find the 3rd term. By carefully reversing the operations, we'll navigate back through the sequence and uncover the value of the 3rd term. This methodical approach is key to solving the problem accurately and efficiently.

Step-by-Step Calculation

Let's put our strategy into action with a step-by-step calculation to find the 3rd term of the sequence. We begin with the 5th term, which is 489. Our goal is to reverse the operations – first dividing by 3 and then adding 2 – to find the preceding terms.

Step 1: Finding the 4th Term

To find the 4th term, we reverse the last operations that were applied to obtain the 5th term. This involves dividing the 5th term by 3 and then adding 2.

  • Divide 489 by 3: 489 / 3 = 163
  • Add 2 to the result: 163 + 2 = 165

Therefore, the 4th term of the sequence is 165.

Step 2: Finding the 3rd Term

Now that we have the 4th term, we can repeat the process to find the 3rd term. We again reverse the operations – dividing by 3 and then adding 2.

  • Divide 165 by 3: 165 / 3 = 55
  • Add 2 to the result: 55 + 2 = 57

Thus, the 3rd term of the sequence is 57. By systematically reversing the operations, we have successfully navigated back through the sequence and uncovered the value of the 3rd term. This step-by-step approach demonstrates how we can solve complex problems by breaking them down into manageable steps.

Verification and Conclusion

To ensure the accuracy of our solution, it's always prudent to verify our findings. We can do this by applying the original term-to-term rule – subtract 2, then multiply by 3 – starting from the 3rd term and checking if we arrive at the 5th term.

  • Starting with the 3rd term (57), subtract 2: 57 - 2 = 55
  • Multiply the result by 3: 55 * 3 = 165 (this is the 4th term)
  • Subtract 2 from the 4th term: 165 - 2 = 163
  • Multiply the result by 3: 163 * 3 = 489 (this is the 5th term)

Since applying the term-to-term rule from the 3rd term leads us to the 5th term, we can confidently conclude that our solution is correct. The 3rd term of the sequence is indeed 57. This verification step reinforces the importance of checking our work to ensure accuracy in mathematical problem-solving.

In conclusion, we have successfully unraveled the sequence and found the 3rd term by carefully reversing the operations defined in the term-to-term rule. This problem highlights the importance of understanding the order of operations and applying a systematic approach to solve complex mathematical challenges. The process of reversing operations is a valuable technique that can be applied to various mathematical problems, making it a fundamental skill for any aspiring mathematician. By mastering these techniques, we can confidently tackle intricate problems and discover elegant solutions.

Key Takeaways

  • Understanding Sequences: Sequences are ordered lists of numbers or mathematical objects, and term-to-term rules define how to move from one term to the next.
  • Reversing Operations: To find previous terms in a sequence when given a later term and the rule, reverse the operations in the rule (undo multiplication before subtraction, for example).
  • Step-by-Step Approach: Break down complex problems into smaller, manageable steps to make the solution process clearer and less error-prone.
  • Verification: Always verify your solution by applying the original rule to check if it leads to the known terms in the sequence.
  • Systematic Thinking: A methodical and systematic approach is crucial for solving mathematical problems accurately and efficiently.

By applying these key takeaways, you can confidently tackle a wide range of sequence-related problems and enhance your mathematical problem-solving skills.