Summary of Arithmetic Progression: Sum

TOPICS

Keywords

• Arithmetic Progression (AP)
• General Term (an)
• Common Difference (r)
• Sum of Terms (Sn)
• Number Sequence
• Initial Term (a1)

Key Questions

• How to identify an AP?
• What is the formula for the general term of an AP?
• How to calculate the sum of terms in an AP?
• What is the relationship between the sum and terms of an AP?

Crucial Topics

• Definition: Numeric sequence where the difference between successive terms is constant.
• General Term Formula: an = a1 + (n - 1) * r
• Sum of Terms Formula: Sn = n/2 * (a1 + an)

Formulas

• General Term of AP: an = a1 + (n - 1) * r
• Sum of n Terms in an AP: Sn = (n * (a1 + an)) / 2 or Sn = n/2 * (2a1 + (n-1) * r)

NOTES

Key Terms

• Arithmetic Progression (AP): Numeric sequence where each term, starting from the second, is equal to the previous term plus a constant r (common difference).
• General Term (an): Value of any term in the sequence, located at position n.
• Common Difference (r): Constant difference between consecutive terms.
• Sum of Terms (Sn): Result of adding the first n terms of the AP.
• Initial Term (a1): First element of the sequence.

Main Ideas, Information, and Concepts

• An AP is determined by its first term and common difference. These define the entire sequence.
• The common difference is the key piece that allows us to find any subsequent term in the sequence.
• The sum of terms in an AP can be calculated without the need to add each term individually.

Topic Contents

• To find the sum of the first n terms (Sn) of an AP, we use one of two equivalent formulas, depending on the available data:
  • First Sum Formula (using the first and last term): Sn = n/2 * (a1 + an)
  • Second Sum Formula (using the first term and the common difference): Sn = n/2 * (2a1 + (n - 1) * r)

Examples and Cases

• Example 1: Given the AP (2, 4, 6, 8, 10), calculate the sum of the first 5 terms.

  • We identify the first term (a1 = 2) and the common difference (r = 2).
  • We use the second sum formula: Sn = 5/2 * (2*2 + (5 - 1) * 2) = Sn = 5/2 * (4 + 8) = Sn = 5/2 * 12 = Sn = 30
  • The sum of terms is 30.

• Example 2: If we have a1 = 3 and common difference r = 5, what is the sum of the first 20 terms?

  • With the first term (a1 = 3) and the common difference (r = 5), we determine the general term of the AP: an = 3 + (20 - 1) * 5 = an = 3 + 95 = an = 98.
  • We use the first sum formula: Sn = 20/2 * (3 + 98) = Sn = 10 * 101 = Sn = 1010
  • The sum of terms is 1010.

These formulas and examples are essential as they allow for quick and efficient calculation of the sum of terms in an AP, a useful skill in various mathematical and everyday applications.

SUMMARY

Key Points

• AP Definition: A sequence of numbers where the difference between each pair of consecutive terms is constant, called the common difference (r).
• General Term (an): Allows to calculate any term of the sequence using the formula an = a1 + (n - 1) * r.
• Sum of Terms (Sn): The sum of the first n terms can be quickly found using the formulas Sn = n/2 * (a1 + an) or Sn = n/2 * (2a1 + (n - 1) * r).

Conclusions

• The structure of an AP is defined by its first term (a1) and the common difference (r).
• The sum of terms in an AP does not require the individual addition of each term, but the application of specific formulas for quick calculation.
• Knowledge of the general term and sum formulas is essential to solve problems involving AP.
• The ability to calculate the sum of an AP has practical applicability in various contexts, reinforcing the importance of understanding and applying such mathematical concepts.