To find the common difference 'd' and the first term 'a' of the given arithmetic sequence, we will use the general formula for the nth term of an arithmetic sequence:
[tex]\[ a_n = a + (n - 1)d \][/tex]
Given:
- The second term (a₂) is -4.
- The seventh term (a₇) is 36.
Let's set up the equations based on these terms.
For the second term (n = 2):
[tex]\[ a_2 = a + (2 - 1)d \][/tex]
[tex]\[ -4 = a + d \][/tex]
[tex]\[ \text{Equation (1):} \quad a + d = -4 \][/tex]
For the seventh term (n = 7):
[tex]\[ a_7 = a + (7 - 1)d \][/tex]
[tex]\[ 36 = a + 6d \][/tex]
[tex]\[ \text{Equation (2):} \quad a + 6d = 36 \][/tex]
Now, we need to solve these two equations simultaneously to find 'a' and 'd'.
First, we'll subtract Equation (1) from Equation (2) to eliminate 'a':
[tex]\[ (a + 6d) - (a + d) = 36 - (-4) \][/tex]
[tex]\[ a + 6d - a - d = 36 + 4 \][/tex]
[tex]\[ 5d = 40 \][/tex]
[tex]\[ d = \frac{40}{5} \][/tex]
[tex]\[ d = 8 \][/tex]
Now that we have the common difference 'd', we can substitute it back into Equation (1) to find 'a':
[tex]\[ a + d = -4 \][/tex]
[tex]\[ a + 8 = -4 \][/tex]
[tex]\[ a = -4 - 8 \][/tex]
[tex]\[ a = -12 \][/tex]
So, the common difference 'd' is 8, and the first term 'a' is -12.
Summary:
- Common difference [tex]\( d = 8 \)[/tex]
- First term [tex]\( a = -12 \)[/tex]
