Question

points on graph 3,1 2,5 1,9
The graph represents an arithmetic sequence. Find the common difference, d, the fourth term, a4
, and the fifth term, a5
, of the arithmetic sequence.

(2 po

Answer 1

To determine the common difference \(d\), the fourth term \(a_4\), and the fifth term \(a_5\) of the arithmetic sequence represented by the points \((3, 1)\), \((2, 5)\), \((1, 9)\), we first need to identify the sequence of \(y\)-values associated with the \(x\)-values in descending order.

The points correspond to the following values:
- For \(x = 1\), \(y = 9\)
- For \(x = 2\), \(y = 5\)
- For \(x = 3\), \(y = 1\)

The \(y\)-values in decreasing order of \(x\) are \(9\), \(5\), and \(1\).

Now, let's find the common difference \(d\):

1. From \(9\) to \(5\), the difference is:
\[
5 - 9 = -4
\]

2. From \(5\) to \(1\), the difference is:
\[
1 - 5 = -4
\]

Since the difference is consistent, the common difference \(d\) is:
\[
d = -4
\]

Next, we need to find the fourth term \(a_4\) and the fifth term \(a_5\).

The first term \(a_1\) (when \(x=1\)) is \(9\).

Using the common difference, we can calculate:
\[
a_2 = a_1 + d = 9 + (-4) = 5
\]
\[
a_3 = a_2 + d = 5 + (-4) = 1
\]
\[
a_4 = a_3 + d = 1 + (-4) = -3
\]
\[
a_5 = a_4 + d = -3 + (-4) = -7
\]

Summarizing the results:
- Common difference \(d = -4\)
- Fourth term \(a_4 = -3\)
- Fifth term \(a_5 = -7\)