Question

1. (10 points) find the first term, (a_1), and the common difference, (d), of the arithmetic sequence given that (a_{48} = 79) and (a_{13} = 9).

1. (10 points) given the arithmetic sequence below, find (n) if (a_n = 97).

-1, 1, 3, 5, 7, dots

Explanation:

Step1: Set up equations for Question 4

$$a_{48} = a_1 + 47d = 79$$

$$a_{13} = a_1 + 12d = 9$$

Step2: Solve for the common difference \(d\)

$$(a_1 + 47d) - (a_1 + 12d) = 79 - 9 \implies 35d = 70 \implies d = 2$$

Step3: Solve for the first term \(a_1\)

$$a_1 + 12(2) = 9 \implies a_1 + 24 = 9 \implies a_1 = -15$$

Step4: Identify sequence parameters for Question 5

$$a_1 = -1, \quad d = 1 - (-1) = 2, \quad a_n = 97$$

Step5: Solve for \(n\) using the formula

$$a_n = a_1 + (n - 1)d \implies 97 = -1 + (n - 1)2$$

$$98 = 2(n - 1) \implies n - 1 = 49 \implies n = 50$$

Answer:

1. \(a_1 = -15\), \(d = 2\)
2. \(n = 50\)