Certainly! To find the [tex]$8^{\text{th}}$[/tex] and [tex]$20^{\text{th}}$[/tex] terms of an arithmetic sequence given the first term and the common difference, we can use the formula for the [tex]$n^{\text{th}}$[/tex] term of an arithmetic sequence:
[tex]\[ a_n = a + (n-1) \cdot d \][/tex]
where:
- [tex]\( a \)[/tex] is the first term,
- [tex]\( d \)[/tex] is the common difference,
- [tex]\( n \)[/tex] is the term number.
Here, the first term [tex]\( a \)[/tex] is 5 and the common difference [tex]\( d \)[/tex] is 3.
### Finding the [tex]$8^{\text{th}}$[/tex] term:
1. Substitute [tex]\( n = 8 \)[/tex], [tex]\( a = 5 \)[/tex], and [tex]\( d = 3 \)[/tex] into the formula:
[tex]\[ a_8 = 5 + (8-1) \cdot 3 \][/tex]
2. Calculate inside the parenthesis first:
[tex]\[ 8-1 = 7 \][/tex]
3. Then multiply by the common difference:
[tex]\[ 7 \cdot 3 = 21 \][/tex]
4. Finally, add to the first term:
[tex]\[ a_8 = 5 + 21 = 26 \][/tex]
Therefore, the [tex]$8^{\text{th}}$[/tex] term is 26.
### Finding the [tex]$20^{\text{th}}$[/tex] term:
1. Substitute [tex]\( n = 20 \)[/tex], [tex]\( a = 5 \)[/tex], and [tex]\( d = 3 \)[/tex] into the formula:
[tex]\[ a_{20} = 5 + (20-1) \cdot 3 \][/tex]
2. Calculate inside the parenthesis first:
[tex]\[ 20-1 = 19 \][/tex]
3. Then multiply by the common difference:
[tex]\[ 19 \cdot 3 = 57 \][/tex]
4. Finally, add to the first term:
[tex]\[ a_{20} = 5 + 57 = 62 \][/tex]
Therefore, the [tex]$20^{\text{th}}$[/tex] term is 62.
In summary:
- The [tex]$8^{\text{th}}$[/tex] term is 26.
- The [tex]$20^{\text{th}}$[/tex] term is 62.
