91Ó°ÊÓ

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.

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 \(n\).

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 \(1 + 2 = 3\).

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.