Question

write an equation for the nth term of the arithmetic sequence.
-7, -4, -1, 2, ...

a) ( a_n = 3n + 10 )

b) ( a_n = 10 + n )

c) ( a_n = 3n - 10 )

d) ( a_n = 3(n - 10) )

part b
graph the first five terms of the sequence.

a) graph with points

b) graph with points

c) graph with points

Explanation:

Part A

Step1: Identify the first term and common difference

The arithmetic sequence is \(-7, -4, -1, 2, \dots\). The first term \(a_1 = -7\). The common difference \(d\) is calculated by subtracting consecutive terms: \(-4 - (-7) = 3\), \(-1 - (-4) = 3\), \(2 - (-1) = 3\), so \(d = 3\).

Step2: Use the formula for the nth term of an arithmetic sequence

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n - 1)d\). Substitute \(a_1 = -7\) and \(d = 3\) into the formula:
\[

$$\begin{align*} a_n &= -7 + (n - 1) \times 3 \\ &= -7 + 3n - 3 \\ &= 3n - 10 \end{align*}$$

\]

First, find the first five terms using the formula \(a_n = 3n - 10\):

• For \(n = 1\): \(a_1 = 3(1) - 10 = -7\)
• For \(n = 2\): \(a_2 = 3(2) - 10 = -4\)
• For \(n = 3\): \(a_3 = 3(3) - 10 = -1\)
• For \(n = 4\): \(a_4 = 3(4) - 10 = 2\)
• For \(n = 5\): \(a_5 = 3(5) - 10 = 5\)

Now, we analyze the graphs:

• The first five terms are \((1, -7)\), \((2, -4)\), \((3, -1)\), \((4, 2)\), \((5, 5)\). These points should show an increasing linear pattern (since the sequence is arithmetic with positive common difference).

Looking at the options:

• Option A: The points seem to match the coordinates \((1, -7)\), \((2, -4)\), \((3, -1)\), \((4, 2)\), \((5, 5)\) (visually, the y-values increase by 3 each time as \(n\) increases by 1).
• Option B: The points are decreasing, which contradicts the positive common difference.
• Option C: The points do not align with the calculated terms.

So the correct graph is Option A.

Answer:

C) \(a_n = 3n - 10\)

Part B