Arithmetic and Geometric Sequences: Concepts and Summation Techniques

Finding the $n$ th 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 $n$ th 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 $100, 20, \dots$ , $r = \frac{20}{100} = \frac{1}{5}$