Lecture Notes: Finding the Sum of the First n Terms of an Arithmetic Sequence

Key Concepts

• Arithmetic Sequence: A sequence of numbers in which the difference between consecutive terms is constant.
• Sum of the First n Terms Formula: [ S_n = \frac{n}{2} \times (2a_1 + (n-1) \cdot d) ] where:
  • ( S_n ) is the sum of the first n terms
  • ( n ) is the number of terms
  • ( a_1 ) is the first term
  • ( d ) is the common difference

Examples

Example 1: Sum of Terms from 5 to 50

• Arithmetic Sequence: 5, 10, 15, 20, ..., 50
• First term, ( a_1 ) = 5
• Last term = 50
• Sum formula: List and add all terms
  • 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50
  • Sum using calculator: 275
  • Therefore, the sum is 275

Example 2: Sum of the First 20 Natural Numbers

• Natural Numbers: 1, 2, 3, ..., 20
• First term, ( a_1 ) = 1
• Last term, ( a_{20} ) = 20
• Number of terms, ( n ) = 20
• Formula: [ S_n = \frac{n}{2} \times (a_1 + a_n) ]
  • [ S_{20} = \frac{20}{2} \times (1 + 20) ]
  • [ S_{20} = 10 \times 21 = 210 ]
  • Therefore, the sum is 210_

Example 3: Sum of the First 16 Terms of the Sequence 8, 11, 14, 17, 20

• Given Sequence: 8, 11, 14, 17, 20
• First term, ( a_1 ) = 8
• Number of terms, ( n ) = 16
• Common difference, ( d ) = 3 (Calculated by: 11 - 8, 14 - 11, etc.)
• Using formula: [ S_n = \frac{n}{2} \times (2a_1 + (n-1) \cdot d) ]
  • [ S_{16} = \frac{16}{2} \times (2 \times 8 + 15 \times 3) ]
  • Substitute and compute:
    • [ S_{16} = 8 \times (16 + 45) ]
    • [ S_{16} = 8 \times 61 = 488 ]
  • Therefore, the sum of the first 16 terms is 488_

Conclusion

• Key to solving these problems is identifying given parameters (first term, number of terms, common difference, last term if applicable) and applying the appropriate formula.