Problem 37 - 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. 14th term is \(-1 ; 18\) th term is -9

Step-by-Step Solution

Answer

First term: 25, Common difference: -2, Recursive formula: \( a_{n} = a_{n-1} - 2 \), nth Term: \( a_{n} = 27 - 2n \)

Step 1: Set Up the Equations

The nth term in an arithmetic sequence can be given by the formula: \[ a_n = a + (n - 1) \times d \]Here, \(a\) is the first term and \(d\) is the common difference. Using this formula, set up equations for the 14th and 18th terms.

Step 2: Write Equations for Given Terms

For the 14th term:\[ a + 13d = -1 \]For the 18th term:\[ a + 17d = -9 \]

Step 3: Solve the System of Equations for a and d

Subtract the first equation from the second equation to eliminate \(a\):\[ (a + 17d) - (a + 13d) = -9 - (-1) \]This simplifies to: \[ 4d = -8 \]Solve for \(d\): \[ d = -2 \]Substitute \(d = -2\) into the first equation to find \(a\):\[ a + 13(-2) = -1 \]\[ a - 26 = -1 \]\[ a = 25 \]

Step 4: Write the Recursive Formula

The recursive formula for an arithmetic sequence is \[ a_{n} = a_{n-1} + d \]Given \(a = 25\) and \(d = -2\), the recursive formula is:\[ a_{1} = 25 \]\[ a_{n} = a_{n-1} - 2 \] for \(n > 1\)

Step 5: Write the Formula for the nth Term

The formula for the nth term is:\[ a_{n} = a + (n-1) \times d \]Substitute \(a = 25\) and \(d = -2\):\[ a_{n} = 25 + (n - 1) \times (-2) \]\[ a_{n} = 25 - 2n + 2 \]Simplify:\[ a_{n} = 27 - 2n \]

Key Concepts

Recursive Formula

A recursive formula helps to find any term in a sequence using the previous term. It's like a step-by-step guide where each term depends on its predecessor. For an arithmetic sequence, the recursive formula is generally written as: \[ a_{n} = a_{n-1} + d \] where \( a_{n} \) is the nth term, \( a_{n-1} \) is the term before it, and \( d \) is the common difference. In our specific sequence, we found that \( d = -2 \) and \( a = 25 \). So, the recursive formula for this sequence is: \[ a_{1} = 25 \] \[ a_{n} = a_{n-1} - 2 \] for \( n > 1 \). Thus, with this formula, you can easily find the value of any term if you know the previous term. For example, if you know \( a_{1} = 25 \), then \( a_{2} = 25 - 2 = 23 \).

Nth term formula

The nth term formula for an arithmetic sequence lets you calculate any term directly without knowing the previous ones. For an arithmetic sequence, this formula is: \[ a_{n} = a + (n-1) \times d \] where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number. In our exercise, we determined \( a = 25 \) and \( d = -2 \). Substituting these values into the formula gives us: \[ a_{n} = 25 + (n-1) \times (-2) \] Simplifying further: \[ a_{n} = 25 - 2n + 2 \] \[ a_{n} = 27 - 2n \] With this formula, you can find any term in the sequence instantly. For example, to find the 10th term: \[ a_{10} = 27 - 2 \times 10 = 27 - 20 = 7 \] Which means \( a_{10} = 7 \).

Common Difference

In an arithmetic sequence, the common difference (\( d \)) is the difference between any two consecutive terms. It remains constant throughout the sequence. To find the common difference, you can subtract the first term from the second term, or any term from the term that follows it. In our example, we used the equations for the 14th and 18th terms to find \( d \). Given: \[ a + 13d = -1 \] \[ a + 17d = -9 \] By subtracting these equations: \[ (a + 17d) - (a + 13d) = -9 - (-1) \] This simplifies to: \[ 4d = -8 \] Solving for \( d \), we get: \[ d = -2 \] This constant difference, \( d = -2 \), tells us that each term in the sequence decreases by 2 as we move from one term to the next. For example, starting from 25, the next few terms would be 23, 21, 19, etc.