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How to Solve an Arithmetic Sequence?

Published in Arithmetic Sequences 3 mins read

Solving an arithmetic sequence involves a few key steps, primarily focusing on identifying the sequence and then applying the appropriate formulas. Here's a breakdown:

Understanding Arithmetic Sequences

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference.

Steps to Solve an Arithmetic Sequence

The process generally involves the following steps, according to the provided reference:

  1. Verification: First, check if the given sequence is indeed an arithmetic sequence. To do this, look for a consistent difference between consecutive terms.
  2. Calculate the Common Difference (d): If the sequence is arithmetic, you must determine the common difference, represented as 'd'. Use the formula:
    d = a₂ - a₁ = a₃ - a₂ = ... = aₙ - aₙ₋₁

    where a₁, a₂, a₃, ..., aₙ are the terms in the sequence.

  3. Solve for Specific Terms or Sums: Once you know 'd', you can:
    • Find any term of the sequence (the nth term) using the formula:
      aₙ = a₁ + (n - 1)d

      where aₙ is the nth term, a₁ is the first term, n is the term number, and d is the common difference.

    • Calculate the sum of the first n terms (Sₙ) using the formula:
        Sₙ = n/2 * (2a₁ + (n - 1)d)

      or

        Sₙ = n/2 * (a₁ + aₙ)

      where Sₙ is the sum of the first n terms.

Example

Let's consider the arithmetic sequence: 2, 5, 8, 11, ...

  1. Verification: The difference between consecutive terms is consistent (5-2=3, 8-5=3, 11-8=3), so it's an arithmetic sequence.
  2. Common Difference (d): d = 5 - 2 = 3
  3. Finding the 10th term (a₁₀):
    • a₁ = 2, n=10, d=3
    • a₁₀ = 2 + (10-1) 3 = 2 + 9 3 = 2 + 27 = 29
  4. Finding the sum of the first 10 terms (S₁₀):
    • S₁₀ = 10/2 (2 2 + (10-1) 3) = 5 (4 + 27) = 5 * 31 = 155
    • Alternatively, you can first find the 10th term (a₁₀), which is 29. So, S₁₀ = 10/2 (2 + 29) = 5 31 = 155

Summary

Step Description Formula
1. Verification Check if the difference between consecutive terms is consistent. N/A
2. Common Difference Calculate the constant difference (d) between terms. d = a₂ - a₁
3. Nth Term Find the value of a specific term in the sequence. aₙ = a₁ + (n - 1)d
4. Sum of n terms Find the sum of the first 'n' terms of the sequence. Sₙ = n/2 * (2a₁ + (n - 1)d)
or Sₙ = n/2 * (a₁ + aₙ)

By following these steps and using the formulas, you can successfully solve and understand various aspects of arithmetic sequences.

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