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. 4th term is \(3 ; 20\) th term is 35

Short answer

First term: \( -3 \), Common difference: \( 2 \), Recursive formula: \( a_n = a_{n-1} + 2 \), nth term formula: \( a_n = 2n - 5 \)

Step-by-step solution

- Identify Given Terms

The 4th term of the arithmetic sequence is given as 3, and the 20th term is given as 35. These are represented as: \[ a_4 = 3 \] \[ a_{20} = 35 \]

- Use the Definition of Arithmetic Sequence

In an arithmetic sequence, the nth term can be expressed as: \[ a_n = a_1 + (n-1)d \] where \( a_1 \) is the first term and \( d \) is the common difference.

- Set Up Equations for Given Terms

Substitute the given terms into the formula: \( a_4 = a_1 + 3d = 3 \) \( a_{20} = a_1 + 19d = 35 \)

- Solve for the Common Difference \( d \)

We have two equations:\[ a_1 + 3d = 3 \] \[ a_1 + 19d = 35 \]Subtract the first equation from the second to eliminate \( a_1 \) and solve for \( d \):\[ (a_1 + 19d) - (a_1 + 3d) = 35 - 3 \] \[ 16d = 32 \] \[ d = 2 \]

- Solve for the First Term \( a_1 \)

Substitute \( d = 2 \) back into one of the original equations to solve for \( a_1 \):\[ a_1 + 3(2) = 3 \] \[ a_1 + 6 = 3 \] \[ a_1 = -3 \]

- Write the Recursive Formula

The recursive formula for an arithmetic sequence is:\[ a_n = a_{n-1} + d \]Substitute the value of \( d \) to get:\[ a_n = a_{n-1} + 2 \]

- Write the General Formula for the nth Term

Substitute the values of \( a_1 \) and \( d \) into the general formula:\[ a_n = a_1 + (n-1)d \] \[ a_n = -3 + (n-1) \times 2 \] Simplify:\[ a_n = -3 + 2n - 2 \] \[ a_n = 2n - 5 \]

Key concepts

Common Difference

The common difference in an arithmetic sequence is the amount by which each term in the sequence increases or decreases from the previous term. It is denoted by the symbol \(d\). To find the common difference, we use the formula: \[ d = a_{n} - a_{n-1} \] In this exercise, the 4th term is \(3\) (\(a_4 = 3\)) and the 20th term is \(35\) (\(a_{20} = 35\)). We set up the following equations using the general nth term formula, \(a_n = a_1 + (n-1)d\): \[ a_4 = a_1 + 3d = 3 \] \[ a_{20} = a_1 + 19d = 35 \] By subtracting these equations to eliminate \(a_1\), we get: \[ (a_1 + 19d) - (a_1 + 3d) = 35 - 3 \] \[ 16d = 32 \] \[ d = 2 \] So, the common difference \(d\) is \(2\). This tells us that each term in the sequence is \(2\) units more than the previous term.

Nth Term Formula

The nth term formula of an arithmetic sequence is used to find any term in the sequence without having to list all the preceding terms. The general form of the nth term formula is: \[ a_n = a_1 + (n-1)d \] Here, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the position of the term in the sequence. According to the solved exercise: \[ a_1 = -3 \] \[ d = 2 \] We substitute these values into the formula to get: \[ a_n = -3 + (n-1) \times 2 \] Simplifying, we get: \[ a_n = -3 + 2n - 2 \] \[ a_n = 2n - 5 \] This formula can now be used to find the value of any term in the sequence by substituting the desired term number \(n\).

Recursive Formula

The recursive formula for an arithmetic sequence provides a way to find any term in the sequence based on the previous term. The general recursive formula is: \[ a_n = a_{n-1} + d \] Using this formula, each term is derived by adding the common difference to the previous term. In the problem exercise, we already know: \[ d = 2 \] Hence, the recursive formula for this specific sequence is: \[ a_n = a_{n-1} + 2 \] It means that once you know the preceding term, simply add \(2\) to find the next term. For example, if the previous term (\(a_{n-1}\)) is \(1\), the next term (\(a_n\)) would be \(1 + 2 = 3\).

Sequence Terms

In arithmetic sequences, each specific value is called a term. Each term is represented by \(a_n\), where \(n\) indicates the term's position in the sequence. Starting with the first term as \(a_1\), subsequent terms are influenced by the common difference \(d\). For example, the sequence discussed in this exercise with first term \(a_1 = -3\) and common difference \(d = 2\):

• The first term is \(-3\)
• The second term, \(a_2 = a_1 + 2 = -1\)
• The third term, \(a_3 = a_2 + 2 = 1\)
• And so forth...

Each term increases by \(2\) from the previous term, forming a sequence: \(-3, -1, 1, 3, 5, \text{...}\). With the explicit nth term formula \[ a_n = 2n - 5 \] , you can directly compute the value of any term without knowing the preceding terms.