In this problem, we need to find the 6th term from the end of the given arithmetic progression (AP). The given AP is: [tex]\( 5, 2, -1, -4, -31 \)[/tex].
Here's how you can find the 6th term from the end step-by-step:
1. Identify the total number of terms in the AP:
The given sequence of terms in the AP includes [tex]\(5\)[/tex] terms: [tex]\(5, 2, -1, -4, -31\)[/tex].
2. Determine the position from the beginning:
To find the 6th term from the end, we need to convert it to the [tex]\(n\)[/tex]-th term from the beginning. If the total number of terms [tex]\(n\)[/tex] is [tex]\(5\)[/tex], then the position we need from the beginning can be found as [tex]\( n - 6 + 1 \)[/tex].
3. Calculate the term position from the start:
- [tex]\( n = 5 \)[/tex]
- [tex]\( 6 \text{th term from the end} \)[/tex] translates to [tex]\( 5 - 6 + 1 = 0 \)[/tex]-th term from the start.
Since [tex]\(0\)[/tex]-th term from the start does not exist, this means there is no valid 6th term from the end in this particular sequence because the total number of terms is less than 6.
So, none of the given answer choices [tex]\( -25, -22, -19, -16 \)[/tex] are correct.
Based on this, there is an apparent mistake in the problem as posed; it's impossible to determine the 6th term from the end for a sequence with only 5 elements.
