Sure! Let's solve this step-by-step.
Given:
- First term of the arithmetic sequence: [tex]\( a_1 = 6 \)[/tex]
- Eighth term of the arithmetic sequence: [tex]\( a_8 = 48 \)[/tex]
### 1. Determine the common difference [tex]\(d\)[/tex]
The formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
For the eighth term ([tex]\( a_8 \)[/tex]):
[tex]\[ a_8 = a_1 + (8 - 1) \cdot d \][/tex]
[tex]\[ 48 = 6 + 7d \][/tex]
Subtract 6 from both sides:
[tex]\[ 42 = 7d \][/tex]
Divide both sides by 7:
[tex]\[ d = 6 \][/tex]
So, the common difference [tex]\(d\)[/tex] is 6.
### 2. Find the 70th term ([tex]\( a_{70} \)[/tex])
Using the formula for the [tex]\(n\)[/tex]-th term again:
[tex]\[ a_{70} = a_1 + (70 - 1) \cdot d \][/tex]
[tex]\[ a_{70} = 6 + 69 \cdot 6 \][/tex]
[tex]\[ a_{70} = 6 + 414 \][/tex]
[tex]\[ a_{70} = 420 \][/tex]
So, the 70th term is 420.
### 3. Calculate the sum of the first 70 terms ([tex]\( S_{70} \)[/tex])
The formula for the sum of the first [tex]\(n\)[/tex] terms of an arithmetic sequence is:
[tex]\[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \][/tex]
For 70 terms:
[tex]\[ S_{70} = \frac{70}{2} \cdot (6 + 420) \][/tex]
[tex]\[ S_{70} = 35 \cdot 426 \][/tex]
[tex]\[ S_{70} = 14910 \][/tex]
So, the sum of the arithmetic sequence up to the 70th term is 14,910.
Thus, the correct answer is:
- 14,910
