To determine the value(s) of [tex]\( m \)[/tex] for the given linear number pattern, we need to ensure that the differences between consecutive terms are constant. Since we have the first three terms given, we can set up the problem step by step as follows:
1. Identify the terms:
- First term: [tex]\( T_1 = m + 1 \)[/tex]
- Second term: [tex]\( T_2 = m^2 + m \)[/tex]
- Third term: [tex]\( T_3 = 3m^2 - m - 4 \)[/tex]
2. Calculate the differences between consecutive terms:
- The difference between the second and the first term:
[tex]\[
\Delta_1 = T_2 - T_1 = (m^2 + m) - (m + 1)
\][/tex]
Simplifying this, we get:
[tex]\[
\Delta_1 = m^2 + m - m - 1 = m^2 - 1
\][/tex]
- The difference between the third and the second term:
[tex]\[
\Delta_2 = T_3 - T_2 = (3m^2 - m - 4) - (m^2 + m)
\][/tex]
Simplifying this, we get:
[tex]\[
\Delta_2 = 3m^2 - m - 4 - m^2 - m = 3m^2 - m^2 - m - m - 4 = 2m^2 - 2m - 4
\][/tex]
3. Set the differences equal to each other:
For the pattern to be linear, [tex]\(\Delta_1\)[/tex] must be equal to [tex]\(\Delta_2\)[/tex]:
[tex]\[
m^2 - 1 = 2m^2 - 2m - 4
\][/tex]
4. Solve the equation for [tex]\( m \)[/tex]:
Rearrange the equation to set it to zero:
[tex]\[
m^2 - 1 = 2m^2 - 2m - 4
\][/tex]
[tex]\[
0 = 2m^2 - m^2 - 2m - 4 + 1
\][/tex]
[tex]\[
0 = m^2 - 2m - 3
\][/tex]
5. Factor the quadratic equation:
The quadratic equation [tex]\( m^2 - 2m - 3 \)[/tex] can be factored as:
[tex]\[
0 = (m - 3)(m + 1)
\][/tex]
6. Find the roots of the equation:
Set each factor equal to zero:
[tex]\[
m - 3 = 0 \quad \Rightarrow \quad m = 3
\][/tex]
[tex]\[
m + 1 = 0 \quad \Rightarrow \quad m = -1
\][/tex]
So, the values of [tex]\( m \)[/tex] that satisfy the conditions for the linear number pattern are:
[tex]\[
m = -1 \quad \text{and} \quad m = 3
\][/tex]
