Question

1.6 exploring arithmetic sequences formative assessment d please show and explain your work use the table below. this shows the values of an arithmetic sequence: n x_n 2 5 3 10 4 15 d) what is the common difference (d) of the sequence x_n? e) what is the value of n_41 d = k = w = a_k = a_w = d(w - k)+a_k f) interpret the meaning of n_4 = 15.

Explanation:

Step1: Find common - difference

The common difference \(d\) of an arithmetic sequence can be found by subtracting consecutive terms. Given \(x_2 = 5\), \(x_3=10\), \(d=x_3 - x_2\).
\[d = 10 - 5=5\]

Step2: Identify values for formula

For part e), assume the formula \(A_w=d(w - k)+a_k\). Let's assume we want to find \(x_4\) (it seems there is a mix - up in the notation in the question, we'll assume we use the arithmetic sequence formula \(x_n=x_1+(n - 1)d\). First, we find \(x_1\). Since \(x_2=x_1 + d\), then \(x_1=x_2 - d=5 - 5 = 0\). The formula for the \(n\)th term of an arithmetic sequence is \(x_n=x_1+(n - 1)d\). Here \(x_1 = 0\) and \(d = 5\), for \(n = 4\), \(x_4=0+(4 - 1)\times5=15\) (which is already given in the table, but if we were to use the general formula). If we assume the formula \(A_w=d(w - k)+a_k\), and we know \(d = 5\), if we take \(k = 2\), \(a_k=x_2 = 5\), \(w = 4\), then \(A_4=5\times(4 - 2)+5=5\times2 + 5=15\).

Step3: Interpret the meaning

For part f), in the context of the arithmetic sequence, when \(n = 4\) (the position in the sequence), the value of the term \(x_4\) (denoted as \(n_4\) in a non - standard way in the question) is 15. It means that the fourth term in the arithmetic sequence has a value of 15.

Answer:

d) \(d = 5\)
e) Assuming the correct interpretation and using the arithmetic - sequence formula \(x_n=x_1+(n - 1)d\) with \(x_1 = 0\) and \(d = 5\), for \(n = 4\), the value is 15 (already given in the table). Using the formula \(A_w=d(w - k)+a_k\) with appropriate values (\(d = 5\), \(k = 2\), \(a_k = 5\), \(w = 4\)) also gives 15.
f) The fourth term of the arithmetic sequence has a value of 15.