Question

1. the table below gives some values for an arithmetic sequence. \

$$\begin{tabular}{c|cccccc} term number $(n)$ & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline $n^{\\text{th}} (a_n)$ & 19 & & & & $-5$ & \\end{tabular}$$

a. complete the table. b. write an equation for $a_n$.

Explanation:

Part (a)

Step 1: Find the common difference \( d \)

We know that in an arithmetic sequence, \( a_n = a_1 + (n - 1)d \). We have \( a_1 = 19 \) and \( a_5=-5 \). Using the formula for \( a_5 \):

$$ a_5=a_1+(5 - 1)d $$

Substitute \( a_1 = 19 \) and \( a_5=-5 \):

$$ -5=19 + 4d $$

Subtract 19 from both sides:

$$ -5-19=4d\implies - 24 = 4d $$

Divide both sides by 4:

$$ d=\frac{-24}{4}=-6 $$

Step 2: Find \( a_2,a_3,a_4,a_6 \)

• For \( a_2 \): \( a_2=a_1 + d=19+( - 6)=13 \)
• For \( a_3 \): \( a_3=a_2 + d=13+( - 6)=7 \)
• For \( a_4 \): \( a_4=a_3 + d=7+( - 6)=1 \)
• For \( a_6 \): \( a_6=a_5 + d=-5+( - 6)=-11 \)

So the completed table is:

Term number (\( n \))	1	2	3	4	5	6

Part (b)

Step 1: Recall the arithmetic sequence formula

The formula for the \( n^{\text{th}} \) term of an arithmetic sequence is \( a_n=a_1+(n - 1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.

Step 2: Substitute \( a_1 \) and \( d \)

We know \( a_1 = 19 \) and \( d=-6 \). Substitute these values into the formula:

$$ a_n=19+(n - 1)(-6) $$

Simplify the right - hand side:

$$ a_n=19-6n + 6=25-6n $$

Part (a) Answer:

The completed table is:

Term number (\( n \))	1	2	3	4	5	6

Part (b) Answer:

\( a_n = 25-6n \)

Answer:

Step 1: Recall the arithmetic sequence formula

The formula for the \( n^{\text{th}} \) term of an arithmetic sequence is \( a_n=a_1+(n - 1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.

Step 2: Substitute \( a_1 \) and \( d \)

We know \( a_1 = 19 \) and \( d=-6 \). Substitute these values into the formula:

$$ a_n=19+(n - 1)(-6) $$

Simplify the right - hand side:

$$ a_n=19-6n + 6=25-6n $$

Part (a) Answer:

The completed table is:

Term number (\( n \))	1	2	3	4	5	6

Part (b) Answer:

\( a_n = 25-6n \)