Arithmetic and Geometric Sequences: Concepts and Summation Techniques

Finding the nth Term
- If the first term (a_1) is 8 and the common difference (d) is 4, find which term is 100.
Summation of Arithmetic Sequences with Common Difference d=1 (Consecutive Integers)
- Calculate the sum of numbers from 1 to 60.
- Calculate the sum of numbers from 15 to 75.
Summation of Arithmetic Sequences with Common Difference d > 1
- Sequence: 5, 9, 13, \dots, 81 (common difference d=4)
  - Find the number of terms (n). Then, calculate the sum of this sequence.
- Sequence: 2, 8, 14, \dots, 92 (common difference d=6)
  - Find the number of terms (n). Then, calculate the sum of this sequence.

Identifying the Common Ratio (r)
- For a sequence like 2, 10, 50, \dots, what is the common ratio r?
- For a sequence like 100, 20, \dots, what is the common ratio r?

Finding the nth Term
- 100 = 8 + (n-1)4
- 92 = (n-1)4
- 23 = n-1
- n = 24
- The 24^{th} term is 100.
Summation of Arithmetic Sequences with Common Difference d=1
- Sum of numbers from 1 to 60:
  - S_{60} = \frac{60}{2}(1 + 60) = 30 \times 61 = 1830
- Sum of numbers from 15 to 75:
  - Number of terms (n) = (75 - 15) + 1 = 61
  - S_{61} = \frac{61}{2}(15 + 75) = \frac{61}{2}(90) = 61 \times 45 = 2745
Summation of Arithmetic Sequences with Common Difference d > 1
- Sequence: 5, 9, 13, \dots, 81 (common difference d=4)
  - 81 = 5 + (n-1)4
  - 76 = (n-1)4
  - 19 = n-1
  - n = 20
  - S_{20} = \frac{20}{2}(5 + 81) = 10 \times 86 = 860
- Sequence: 2, 8, 14, \dots, 92 (common difference d=6)
  - 92 = 2 + (n-1)6
  - 90 = (n-1)6
  - 15 = n-1
  - n = 16
  - S_{16} = \frac{16}{2}(2 + 92) = 8 \times 94 = 752
Identifying the Common Ratio (r)
- For 2, 10, 50, \dots, r = \frac{10}{2} = 5
- For 100, 20, \dots, r = \frac{20}{100} = \frac{1}{5}