Verified Answer :
- The sequence is defined by the first term \(a = 3\) and common difference \(d = 4\), yielding the formula \(a_n = 4n - 1\).
- The \(12^{th}\) term is found by substituting \(n=12\), resulting in \(a_{12} = 47\).
- To check if \(95\) is in the sequence, we solve \(4n - 1 = 95\), which gives \(n = 24\), confirming it is the \(24^{th}\) term.
- Final answer: The \(12^{th}\) term is \(\boxed{47}\) and \(95\) is a member of the sequence.
Explanation
- Analysis of the Arithmetic Sequence
The arithmetic sequence provided is \(3, 7, 11, 15, \dots\). We identify the first term as \(a = 3\) and the common difference \(d\) as the difference between any two consecutive terms. Thus, \(d = 7 - 3 = 4\). The general formula for the \(n^{th}\) term of an arithmetic sequence is given by \(a_n = a + (n - 1)d\). Substituting the known values, we get \(a_n = 3 + (n - 1)4 = 3 + 4n - 4 = 4n - 1\). This formula allows us to calculate any term in the sequence by knowing its position \(n\).
- Finding the 12th Term
To find the \(12^{th}\) term of the sequence, we substitute \(n = 12\) into the general formula \(a_n = 4n - 1\):
\[a_{12} = 4(12) - 1 = 48 - 1 = 47.\]
Thus, the \(12^{th}\) term of the sequence is \(47\).
- Checking Membership of 95 and Addressing Question 5
To determine if \(95\) is a member of the sequence, we set the general term \(a_n\) equal to \(95\) and solve for \(n\):
\[4n - 1 = 95\]
Adding \(1\) to both sides:
\[4n = 96\]
Dividing by \(4\):
\[n = \frac{96}{4} = 24.\]
Since \(n = 24\) is a positive integer, it follows that \(95\) is indeed a member of the sequence, specifically the \(24^{th}\) term. Note: As Question 5 was left blank in the request, no further calculations can be performed.
Examples
Arithmetic sequences are used in many real-life situations involving predictable, repeating patterns. For example, consider a savings account where you deposit a fixed amount of money every month: if you start with an initial deposit (the first term) and add the same amount (the common difference) each time, the total amount in your account forms an arithmetic sequence. By using the general term formula, you can quickly project your future balance after any number of months without calculating each intermediate step.