3.1 If and are the first three terms of a linear sequence,2𝑥 − 1, 𝑥 + 5 4𝑥 − 1determine the following:3.1.1 the value of 𝑥. (2)3.1.2 the first three terms. (1)3.1.3 the general rule of the term. (1)3.2 Given the quadratic pattern: 0, − 3, − 10, − 21, − 36...3.2.1 Determine the formula for the term of the pattern.𝑛𝑡ℎ (6)3.2.2 Which term is equal − 820? (5)[15

3.1.1 To find the value of x, we need to set the second term (x + 5) equal to the average of the first and third terms. This gives us the equation:

(x + 5) = [(2x - 1) + (4x - 1)] / 2

Solving this equation gives us x = 3.

3.1.2 Substituting x = 3 into the expressions for the first three terms gives us the first term as 2(3) - 1 = 5, the second term as 3 + 5 = 8, and the third term as 4(3) - 1 = 11. So, the first three terms are 5, 8, and 11.

3.1.3 The general rule of the term in a linear sequence is given by the formula a + (n - 1)d, where a is the first term and d is the common difference. Here, a = 5 and d = 8 - 5 = 3. So, the general rule of the term is 5 + (n - 1)3.

3.2.1 The formula for the nth term of a quadratic sequence can be found by taking differences until a constant difference is found. The second differences here are constant (-7), so the formula is of the form an^2 + bn + c. Substituting the first few terms into this formula and solving the resulting system of equations gives a = -2, b = 1, and c = 0. So, the formula for the nth term is -2n^2 + n.

3.2.2 To find which term is equal to -820, we set the formula for the nth term equal to -820 and solve for n. This gives us the equation -2n^2 + n = -820. Solving this equation gives n = 41. So, the 41st term is -820.