Lecture Notes: Finding the Sum of an Arithmetic Series

Introduction

• Focus on calculating the sum of an arithmetic series using a formula.
• Given series: 4, 7, 10, ..., 58.
• Alternative to manually adding: Use a formula for efficiency.

Formula for Sum of Arithmetic Series

• Formula: ( S_n = \frac{(a_1 + a_n)}{2} \times n )
  • ( a_1 ): First term.
  • ( a_n ): Last term.
  • ( n ): Number of terms.
• For the given series:
  • ( a_1 = 4 )
  • ( a_n = 58 )

Calculating Number of Terms (n)

• Use the formula: ( a_n = a_1 + (n - 1) \times d )
  • d: Common difference between terms.
  • For this series, d = 3 (difference between consecutive terms).
• Steps:
  • ( 58 = 4 + (n - 1) \times 3 )
  • Solve: ( 58 = 3n + 1 )
  • Simplify: ( 57 = 3n )
  • Result: ( n = 19 )

Calculating the Sum

• Apply the sum formula:
  • ( S_{19} = \frac{(4 + 58)}{2} \times 19 )
  • Calculate: 62 / 2 = 31
  • Sum: ( 31 \times 19 = 589 )_

Example 2: Decreasing Arithmetic Series

• Series from 288 to 16, decreasing by 4.
• ( a_1 = 288 ), ( a_n = 16 ), d = -4.
• Find ( n ):
  • ( 16 = 288 + (n - 1) \times (-4) )
  • Solve: ( -276 = -4n )
  • Result: ( n = 69 )

Calculating Sum for Example 2

• Use the sum formula:
  • ( S_{69} = \frac{(288 + 16)}{2} \times 69 )
  • Average: 152
  • Sum: ( 152 \times 69 = 10,488 )_

Example 3: Series Calculation

• Series: 96, 89, 82, 75, ..., 12.
• ( a_1 = 96 ), ( a_n = 12 ), d = -7
• Calculate ( n ):
  • ( 12 = 96 + (n - 1) \times (-7) )
  • Solve: ( -91 = -7n )
  • Result: ( n = 13 )

Sum for Example 3

• Formula application:
  • ( S_{13} = \frac{(96 + 12)}{2} \times 13 )
  • Average: 54
  • Sum: ( 54 \times 13 = 702 )_

Summary

• Key steps for calculating an arithmetic series:
  • Determine number of terms (( n )) using ( a_n = a_1 + (n-1) \times d ).
  • Use ( S_n = \frac{(a_1 + a_n)}{2} \times n ) to find the sum.
• Practice additional problems for mastery.