Problem 36 - Algebra and Trigonometry

Question

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 5th term is \(-2 ; 13\) th term is 30

Step-by-Step Solution

Answer

The first term is \(a = -18\) and the common difference is \(d = 4\). Recursive formula: \(a_n = a_{n-1} + 4\); nth term formula: \(a_n = 4n - 22\)

Step 1 - Identify the given terms

The 5th term of the arithmetic sequence is given as \(-2\), and the 13th term is given as 30.

Step 2 - Set up the n-th term formula

The n-th term of an arithmetic sequence can be written as \(a_n = a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference.

Step 3 - Write equations for the given terms

Use the n-th term formula for the given terms: \(a + 4d = -2\) for the 5th term, and \(a + 12d = 30\) for the 13th term.

Step 4 - Solve the system of equations

Subtract the first equation from the second to eliminate \(a\): \[(a + 12d) - (a + 4d) = 30 - (-2)\]\[8d = 32\]\[d = 4\]

Step 5 - Determine the first term \(a\)

Substitute \(d=4\) back into one of the original equations: \[a + 4 \cdot 4 = -2\]\[a + 16 = -2\]\[a = -18\]

Step 6 - Write the recursive formula

The recursive formula can be written as: \(a_1 = -18\)\(a_n = a_{n-1} + 4\) for \(n > 1\)

Step 7 - Write the explicit formula for the n-th term

Using the n-th term formula and substituting the values of \(a\) and \(d\): \[a_n = -18 + (n-1) \cdot 4\]\[a_n = -18 + 4n - 4\]\[a_n = 4n - 22\]

Key Concepts

First Term

In any arithmetic sequence, the first term is the initial value from which the sequence starts. It’s denoted as \(a_1\). For example, in the sequence described in this exercise, the first term is found by applying the information given about the 5th and 13th terms. We know that \(a_5 = -2\) and \(a_{13} = 30\). Using the nth term formula \(a_n = a + (n-1)d\), we set up two equations:

• \(a + 4d = -2\)
• \(a + 12d = 30\)

Solving the equations simultaneously, we discovered the first term as \(a = -18\). This is the starting point of our sequence.

Common Difference

The common difference in an arithmetic sequence is the consistent difference between consecutive terms. It is denoted as \(d\). To calculate it, you subtract a term from the next term. From the given terms in the exercise, we use the equations from Step 4:

• \(a + 4d = -2\)
• \(a + 12d = 30\)

By subtracting these equations, we find: \[8d = 32\]This simplifies to \[d = 4\]So, the common difference in this sequence is 4, indicating that each term is 4 more than the previous one.

Recursive Formula

The recursive formula for an arithmetic sequence defines each term based on the previous term. It's written as \(a_n = a_{n-1} + d\).In this problem, since we identified that the first term \(a_1\) is -18 and the common difference \(d\) is 4, the recursive formula can be expressed as follows:

• Initial term: \(a_1 = -18\)
• Recursive step: \(a_n = a_{n-1} + 4\), for \(n > 1\)

This means that to find any term, you add 4 to the previous term.

nth Term Formula

The nth term formula of an arithmetic sequence allows you to calculate any term directly. It is expressed as \(a_n = a + (n-1)d\).Given that we've determined \(a = -18\) and \(d = 4\), substitute these values into the formula:\[a_n = -18 + (n-1) \times 4\]This simplifies to:\[a_n = 4n - 22\]Thus, the nth term formula \(a_n = 4n - 22\) allows you to find any term in the sequence directly by plugging in the term number \(n\). For example, if you want the 10th term, just substitute \(n = 10\):\[a_{10} = 4 \times 10 - 22 = 18\]