To find the value of the 13th term in the sequence -7, -3, 1, 5, ..., we can observe that this sequence is arithmetic. In an arithmetic sequence, the difference between consecutive terms remains constant. This difference is known as the common difference (d).
First, let's identify the terms in the sequence and find the common difference:
- First term (a₁) = -7
- Second term (a₂) = -3
- Third term (a₃) = 1
- Fourth term (a₄) = 5
We can see that the common difference (d) between these terms is:
[tex]\[ d = a₂ - a₁ = -3 - (-7) = -3 + 7 = 4 \][/tex]
We need to find the 13th term (a₁₃) of this sequence. The general formula for the nth term of an arithmetic sequence is:
[tex]\[ a_n = a₁ + (n-1) \cdot d \][/tex]
Here, [tex]\( a₁ = -7 \)[/tex], [tex]\( d = 4 \)[/tex], and [tex]\( n = 13 \)[/tex]. Plugging these values into the formula gives:
[tex]\[ a₁₃ = a₁ + (13-1) \cdot d \][/tex]
[tex]\[ a₁₃ = -7 + 12 \cdot 4 \][/tex]
[tex]\[ a₁₃ = -7 + 48 \][/tex]
[tex]\[ a₁₃ = 41 \][/tex]
Therefore, the value of the 13th term in the sequence is 41.
