Answer :
To derive the explicit and recursive formulas for the sequence \( a_n = 2n + 10 \) for \( n \geq 1 \), let's follow a detailed, step-by-step process.
### Explicit Formula
The explicit formula for the sequence is already given as:
[tex]\[ a_n = 2n + 10 \][/tex]
This means that for each term \( n \), you can directly calculate its value by substituting \( n \) into the formula.
### Calculating the First Few Terms
Let's calculate the first few terms to understand the sequence better:
- For \( n = 1 \):
[tex]\[ a_1 = 2(1) + 10 = 2 + 10 = 12 \][/tex]
- For \( n = 2 \):
[tex]\[ a_2 = 2(2) + 10 = 4 + 10 = 14 \][/tex]
### Difference Between Consecutive Terms
To find the recursive formula, we need to determine the common difference between consecutive terms:
[tex]\[ \text{Difference} = a_2 - a_1 \][/tex]
[tex]\[ \text{Difference} = 14 - 12 = 2 \][/tex]
### Recursive Formula
With the difference known, we can now define the recursive formula. The recursive formula is written in terms of a previous term and the common difference:
[tex]\[ a_n = a_{n-1} + 2 \][/tex]
### Initial Term
The initial term of the sequence is \( a_1 \):
[tex]\[ a_1 = 12 \][/tex]
### Summarizing the Recursive Formula
Combining these pieces, the recursive formula for the sequence along with its initial term is:
[tex]\[ a_1 = 12 \][/tex]
[tex]\[ a_n = a_{n-1} + 2 \text{ for } n \geq 2 \][/tex]
### Final Answer
Therefore, the explicit formula is:
[tex]\[ a_n = 2n + 10 \][/tex]
And the recursive formula is:
[tex]\[ a_1 = 12 \][/tex]
[tex]\[ a_n = a_{n-1} + 2 \text{ for } n \geq 2 \][/tex]
These formulations fully describe the sequence both explicitly and recursively.
### Explicit Formula
The explicit formula for the sequence is already given as:
[tex]\[ a_n = 2n + 10 \][/tex]
This means that for each term \( n \), you can directly calculate its value by substituting \( n \) into the formula.
### Calculating the First Few Terms
Let's calculate the first few terms to understand the sequence better:
- For \( n = 1 \):
[tex]\[ a_1 = 2(1) + 10 = 2 + 10 = 12 \][/tex]
- For \( n = 2 \):
[tex]\[ a_2 = 2(2) + 10 = 4 + 10 = 14 \][/tex]
### Difference Between Consecutive Terms
To find the recursive formula, we need to determine the common difference between consecutive terms:
[tex]\[ \text{Difference} = a_2 - a_1 \][/tex]
[tex]\[ \text{Difference} = 14 - 12 = 2 \][/tex]
### Recursive Formula
With the difference known, we can now define the recursive formula. The recursive formula is written in terms of a previous term and the common difference:
[tex]\[ a_n = a_{n-1} + 2 \][/tex]
### Initial Term
The initial term of the sequence is \( a_1 \):
[tex]\[ a_1 = 12 \][/tex]
### Summarizing the Recursive Formula
Combining these pieces, the recursive formula for the sequence along with its initial term is:
[tex]\[ a_1 = 12 \][/tex]
[tex]\[ a_n = a_{n-1} + 2 \text{ for } n \geq 2 \][/tex]
### Final Answer
Therefore, the explicit formula is:
[tex]\[ a_n = 2n + 10 \][/tex]
And the recursive formula is:
[tex]\[ a_1 = 12 \][/tex]
[tex]\[ a_n = a_{n-1} + 2 \text{ for } n \geq 2 \][/tex]
These formulations fully describe the sequence both explicitly and recursively.
