Rodriquez Best
01/25/2024 · Junior High School
3.2 A linear number pattern with a constant difference can be represented by the terms \( x+3 ; 3 x+2 ; 6 x-1 \). Determine the numerical value \( x \) AND the numerical value the \( 3^{\text {rd }} \) term.
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Step-by-step Solution
To determine the value of \( x \) in the linear number pattern represented by the terms \( x + 3 \), \( 3x + 2 \), and \( 6x - 1 \), we need to ensure that the difference between consecutive terms is constant.
Let's denote the terms as follows:
- First term: \( a_1 = x + 3 \)
- Second term: \( a_2 = 3x + 2 \)
- Third term: \( a_3 = 6x - 1 \)
The difference between the second and first terms should equal the difference between the third and second terms:
\[
a_2 - a_1 = a_3 - a_2
\]
Calculating \( a_2 - a_1 \):
\[
a_2 - a_1 = (3x + 2) - (x + 3) = 3x + 2 - x - 3 = 2x - 1
\]
Calculating \( a_3 - a_2 \):
\[
a_3 - a_2 = (6x - 1) - (3x + 2) = 6x - 1 - 3x - 2 = 3x - 3
\]
Setting the two differences equal to each other:
\[
2x - 1 = 3x - 3
\]
Now, we can solve for \( x \):
\[
2x - 1 = 3x - 3
\]
\[
-1 + 3 = 3x - 2x
\]
\[
2 = x
\]
Now that we have found \( x = 2 \), we can substitute this value back into the expression for the third term \( a_3 \):
\[
a_3 = 6x - 1 = 6(2) - 1 = 12 - 1 = 11
\]
Thus, the numerical value of \( x \) is \( 2 \) and the numerical value of the third term is \( 11 \).
Final answers:
- \( x = 2 \)
- Third term \( a_3 = 11 \)
Quick Answer
\( x = 2 \), third term is 11
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