Sure, let's solve this step-by-step. Given the first term ([tex]\( a_1 \)[/tex]) of an arithmetic progression (A.P.) is -5 and the sixth term ([tex]\( a_6 \)[/tex]) is 23, we are to find the common difference ([tex]\( d \)[/tex]).
The general form of the [tex]\( n \)[/tex]-th term of an A.P. is given by:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
Where:
- [tex]\( a_n \)[/tex] is the [tex]\( n \)[/tex]-th term,
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( d \)[/tex] is the common difference,
- [tex]\( n \)[/tex] is the term number.
For the sixth term ([tex]\( a_6 \)[/tex]), we can write:
[tex]\[ a_6 = a_1 + (6-1)d \][/tex]
[tex]\[ 23 = -5 + 5d \][/tex]
Now, let's solve this equation step-by-step to find [tex]\( d \)[/tex]:
1. Start with the equation:
[tex]\[ 23 = -5 + 5d \][/tex]
2. Add 5 to both sides to isolate the term containing [tex]\( d \)[/tex]:
[tex]\[ 23 + 5 = 5d \][/tex]
[tex]\[ 28 = 5d \][/tex]
3. Divide both sides by 5 to solve for [tex]\( d \)[/tex]:
[tex]\[ d = \frac{28}{5} \][/tex]
Therefore, the common difference [tex]\( d \)[/tex] is:
[tex]\[ d = 5.6 \][/tex]
So, the common difference in this arithmetic progression is [tex]\( 5.6 \)[/tex].
