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How to Solve Arithmetic Progression?

Published in Arithmetic Progression 3 mins read

To solve arithmetic progression (AP) problems, you'll need to understand and apply a few key formulas and concepts. An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference.

Key Concepts

Here's a breakdown of how to approach solving AP problems, using the formulas you provided:

  • Common Difference (d): This is the constant value added to each term to get the next term. It's calculated using the formula:
    d = aₙ - aₙ₋₁ , where aₙ is any term and aₙ₋₁ is the term immediately preceding it.

  • nth Term (aₙ): To find any specific term in the sequence, you can use the formula:
    aₙ = a + (n - 1)d
    Where:

    • a is the first term of the sequence.
    • n is the position of the term you want to find.
    • d is the common difference.

  • Sum of n Terms (Sₙ): To calculate the sum of the first 'n' terms in the sequence, use:
    Sₙ = n/2 (2a + (n - 1)d)
    Where:

    • n is the number of terms.
    • a is the first term.
    • d is the common difference.

Steps to Solve AP Problems

Here’s a step-by-step approach using examples:

  1. Identify the given information: Note the values that are provided, such as the first term (a), the common difference (d), a specific term (aₙ), the position of a term (n), or the sum of terms (Sₙ).
  2. Determine what you need to find: Identify the unknown value you want to calculate. It could be d, aₙ, or Sₙ.
  3. Choose the appropriate formula: Select the formula that best relates the known and unknown values:
    • To find 'd': d = aₙ - aₙ₋₁
    • To find 'aₙ': aₙ = a + (n - 1)d
    • To find 'Sₙ': Sₙ = n/2 (2a + (n - 1)d)
  4. Substitute the known values: Plug the known values into the chosen formula.
  5. Solve for the unknown: Perform the necessary arithmetic operations to find the value of the unknown variable.

Examples

Example 1: Finding the nth term

Given an AP with a first term of 2 (a = 2) and a common difference of 3 (d = 3), find the 10th term (a₁₀).

  • Using the formula: aₙ = a + (n - 1)d
  • Substitute: a₁₀ = 2 + (10 - 1) * 3
  • Solve: a₁₀ = 2 + 9 * 3 = 2 + 27 = 29
  • Therefore, the 10th term is 29.

Example 2: Finding the sum of n terms

Given the same AP from Example 1, find the sum of the first 10 terms (S₁₀).

  • Using the formula: Sₙ = n/2 (2a + (n - 1)d)
  • Substitute: S₁₀ = 10/2 (2 * 2 + (10 - 1) * 3)
  • Solve: S₁₀ = 5 (4 + 9 * 3) = 5 (4 + 27) = 5 * 31 = 155
  • Therefore, the sum of the first 10 terms is 155.

Summary Table

Formula Purpose Variables
d = aₙ - aₙ₋₁ Calculates the common difference d, aₙ, aₙ₋₁
aₙ = a + (n - 1)d Calculates the nth term of the sequence aₙ, a, n, d
Sₙ = n/2 (2a + (n - 1)d) Calculates the sum of the first n terms Sₙ, n, a, d

By understanding these formulas and the steps involved, you can effectively solve a wide variety of arithmetic progression problems.

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