An arithmetic sequence has these terms: a Subscript 2 Baseline equals 13, a Subscript 4 Baseline equals 25, a Subscript 6 Baseline equals 37, and a Subscript 8 Baseline equals 49.

Question

Which formula can be used to find a Subscript n Baseline?

Answer options with 5 options
A.	
a Subscript n Baseline equals 13 plus 12 left-parenthesis n minus 1 right-parenthesis

B.	
a Subscript n Baseline equals 13 plus 6 left-parenthesis n minus 1 right-parenthesis

C.	
a Subscript n Baseline equals 7 plus 12 left-parenthesis n minus 1 right-parenthesis

D.	
a Subscript n Baseline equals 7 plus 6 left-parenthesis n minus 1 right-parenthesis

E.	
a Subscript n Baseline equals 6 plus 7 left-parenthesis n minus 1 right-parenthesis

GPT-4o mini · Answer

To find the formula for the $n$-th term $a_n$ of the given arithmetic sequence, we first need to identify the common difference and the first term of the sequence. We have the terms: - $a_2 = 13$ - $a_4 = 25$ - $a_6 = 37$ - $a_8 = 49$ Let's calculate the common difference $d$ using the first two terms we have: 1. $a_4 - a_2 = 25 - 13 = 12$ - Since $a_4$ is the 4th term and $a_2$ is the 2nd term, two terms apart indicates that the difference will be $d \cdot 2 = 12$. Hence, $d = 12/2 = 6$. Now that we have the common difference $d = 6$, we can determine a formula for the $n$-th term. The general formula for an arithmetic sequence is: $$ a_n = a_1 + (n-1)d $$ We already know $d = 6$. We need to find $a_1$, the first term of the sequence. We can find $a_1$ by using $a_2$: $$ a_2 = a_1 + (2-1)d \implies 13 = a_1 + 1 \cdot 6 \implies a_1 + 6 = 13 \implies a_1 = 7 $$ Now we can substitute $a_1$ and $d$ into the formula: $$ a_n = 7 + (n-1)6 $$ Now simplifying this gives: $$ a_n = 7 + 6(n-1) \implies a_n = 7 + 6n - 6 \implies a_n = 6n + 1 $$ So the formula to find $a_n$ is: $$ a_n = 7 + 6(n - 1) $$ Looking at the provided options, we see that option D matches our derived formula. **The correct answer is:** **D. $a_n = 7 + 6(n - 1)$**