Certainly! Let's break down and solve each part of the problem step-by-step.
### Part 1: Finding the first 4 terms for each given nth term
Given the nth term formulas, we need to find the values for [tex]\( n = 1, 2, 3, 4 \)[/tex].
#### a) [tex]\( a_n = n + 4 \)[/tex]
Let’s calculate each term:
- When [tex]\( n = 1 \)[/tex]: [tex]\( a_1 = 1 + 4 = 5 \)[/tex]
- When [tex]\( n = 2 \)[/tex]: [tex]\( a_2 = 2 + 4 = 6 \)[/tex]
- When [tex]\( n = 3 \)[/tex]: [tex]\( a_3 = 3 + 4 = 7 \)[/tex]
- When [tex]\( n = 4 \)[/tex]: [tex]\( a_4 = 4 + 4 = 8 \)[/tex]
So, the first 4 terms of this sequence are: [tex]\(5, 6, 7, 8 \)[/tex].
#### b) [tex]\( a_n = 12 - 3n \)[/tex]
Let’s calculate each term:
- When [tex]\( n = 1 \)[/tex]: [tex]\( a_1 = 12 - 3 \cdot 1 = 12 - 3 = 9 \)[/tex]
- When [tex]\( n = 2 \)[/tex]: [tex]\( a_2 = 12 - 3 \cdot 2 = 12 - 6 = 6 \)[/tex]
- When [tex]\( n = 3 \)[/tex]: [tex]\( a_3 = 12 - 3 \cdot 3 = 12 - 9 = 3 \)[/tex]
- When [tex]\( n = 4 \)[/tex]: [tex]\( a_4 = 12 - 3 \cdot 4 = 12 - 12 = 0 \)[/tex]
So, the first 4 terms of this sequence are: [tex]\(9, 6, 3, 0\)[/tex].
### Part 2: Finding the nth term of the sequence 5, 9, 13, 17, 21, ...
Given sequence: [tex]\(5, 9, 13, 17, 21, \ldots\)[/tex]
This is an arithmetic sequence. In an arithmetic sequence, the nth term is determined using the formula:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
where:
- [tex]\( a_1 \)[/tex] is the first term
- [tex]\( d \)[/tex] is the common difference between consecutive terms
- [tex]\( n \)[/tex] is the term number
From the given sequence:
- The first term [tex]\( a_1 = 5 \)[/tex]
- The common difference [tex]\( d \)[/tex] between consecutive terms is [tex]\( 9 - 5 = 4 \)[/tex]
Substitute these values into the formula:
[tex]\[ a_n = 5 + (n - 1) \cdot 4 \][/tex]
[tex]\[ a_n = 5 + 4n - 4 \][/tex]
[tex]\[ a_n = 4n + 1 \][/tex]
Thus, the nth term of the sequence [tex]\(5, 9, 13, 17, 21, \ldots\)[/tex] is:
[tex]\[ a_n = 4n + 1 \][/tex]
### Summary
1. The first 4 terms of [tex]\( a_n = n + 4 \)[/tex] are [tex]\(5, 6, 7, 8\)[/tex].
2. The first 4 terms of [tex]\( a_n = 12 - 3n \)[/tex] are [tex]\(9, 6, 3, 0\)[/tex].
3. The nth term for the sequence [tex]\(5, 9, 13, 17, 21, \ldots\)[/tex] is [tex]\( a_n = 4n + 1 \)[/tex].
