Question

1. (0 points) find the first term, (a_1), and the common difference, (d), of the arithmetic sequence given that (a_{30} = -147) and (a_{41} = -202).

1. (0 points) given the arithmetic sequence below, find (n) if (a_n = 172).

-12, -8, -4, 0, 4, \dots

Explanation:

Step1: Find the common difference \(d\)

$$d = a_{41} - a_{40} = -202 - (-147) = -55$$

Wait, let me re-read the subscripts in the image. The subscripts are \(a_{30} = -147\) and \(a_{41} = -202\). Let's calculate with these values:

$$a_{41} - a_{30} = 11d \implies -202 - (-147) = -55 \implies 11d = -55 \implies d = -5$$

Step2: Find the first term \(a_1\)

$$a_{30} = a_1 + 29d \implies -147 = a_1 + 29(-5) \implies -147 = a_1 - 145 \implies a_1 = -2$$

Step3: Identify parameters for question 23

$$a_1 = -12, \quad d = -8 - (-12) = 4, \quad a_n = 172$$

Step4: Solve for \(n\)

$$a_n = a_1 + (n-1)d \implies 172 = -12 + (n-1)4 \implies 184 = 4(n-1) \implies n-1 = 46 \implies n = 47$$

Answer:

1. \(a_1 = -2\), \(d = -5\)
2. \(n = 47\)