In an arithmetic progression (AP), the \( n \)-th term can be calculated using the formula:
\[ a_n = a + (n-1)d \]
where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.
Given:
- The 43rd term \( a_{43} = 26 \)
- The common difference \( d = 13.5 \)
We can substitute these values into the formula for the 43rd term:
\[ a_{43} = a + (43 - 1) \cdot d \] \[ 26 = a + 42 \cdot 13.5 \]
Calculating \( 42 \cdot 13.5 \):
\[ 42 \cdot 13.5 = 567 \]
Now substituting this back into the equation:
\[ 26 = a + 567 \]
To find \( a \):
\[ a = 26 - 567 \] \[ a = 26 - 567 = -541 \]
Now we have found the first term \( a = -541 \).
Next, we want to find the 5th term \( a_5 \):
\[ a_5 = a + (5 - 1) \cdot d \] \[ a_5 = -541 + 4 \cdot 13.5 \]
Calculating \( 4 \cdot 13.5 \):
\[ 4 \cdot 13.5 = 54 \]
Now substituting this back into the equation:
\[ a_5 = -541 + 54 \] \[ a_5 = -541 + 54 = -487 \]
Thus, the first term of the progression is \( -541 \) and the 5th term is \( -487 \).