Key Concepts and Formulas

• Arithmetic Progression (AP): A sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the common difference, denoted by $d$.
• nth Term of an AP: The $n^{th}$ term of an AP is given by the formula $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.
• nth Term from the End of an AP: The $n^{th}$ term from the end of an AP is equivalent to the $(N-n+1)^{th}$ term from the beginning, where $N$ is the total number of terms. Alternatively, it can be found by reversing the AP and then finding the $n^{th}$ term of the reversed sequence.

Step-by-Step Solution

Step 1: Identify the given AP and convert terms to fractions. The given progression is $20, 19\frac{1}{4}, 18\frac{1}{2}, 17\frac{3}{4}, \ldots, -129\frac{1}{4}$. To facilitate calculations, we convert the mixed numbers into improper fractions:

• $20 = \frac{20}{1} = \frac{80}{4}$
• $19\frac{1}{4} = \frac{19 \times 4 + 1}{4} = \frac{77}{4}$
• $18\frac{1}{2} = \frac{18 \times 2 + 1}{2} = \frac{37}{2} = \frac{74}{4}$
• $17\frac{3}{4} = \frac{17 \times 4 + 3}{4} = \frac{71}{4}$
• $-129\frac{1}{4} = -\frac{129 \times 4 + 1}{4} = -\frac{517}{4}$ The AP can be written as: $\frac{80}{4}, \frac{77}{4}, \frac{74}{4}, \frac{71}{4}, \ldots, -\frac{517}{4}$.

Step 2: Determine the common difference ($d$) of the original AP. The common difference is the difference between any term and its preceding term. Using the first two terms: $d = \frac{77}{4} - \frac{80}{4} = \frac{77 - 80}{4} = -\frac{3}{4}$.

Step 3: Reverse the AP and determine its properties. To find the $20^{th}$ term from the end, we can reverse the sequence. The last term of the original AP becomes the first term of the reversed AP, and the common difference of the reversed AP is the negative of the original common difference. The reversed AP starts with $-\frac{517}{4}$. The common difference of the reversed AP, let's call it $d'$, is: $d' = -d = -(-\frac{3}{4}) = \frac{3}{4}$.

Step 4: Find the $20^{th}$ term of the reversed AP. Now, we need to find the $20^{th}$ term of the reversed AP. The first term of the reversed AP is $a_{rev} = -\frac{517}{4}$. The common difference of the reversed AP is $d' = \frac{3}{4}$. We want to find the $20^{th}$ term, so $n = 20$. Using the formula $a_n = a + (n-1)d$: $a_{20, rev} = a_{rev} + (20-1)d'$ $a_{20, rev} = -\frac{517}{4} + (19)\left(\frac{3}{4}\right)$ $a_{20, rev} = -\frac{517}{4} + \frac{57}{4}$ $a_{20, rev} = \frac{-517 + 57}{4}$ $a_{20, rev} = \frac{-460}{4}$ $a_{20, rev} = -115$.

This $20^{th}$ term of the reversed AP is the $20^{th}$ term from the end of the original AP.

Common Mistakes & Tips

• Fraction Conversion Errors: Ensure accurate conversion of mixed numbers to improper fractions. Double-check calculations involving addition and subtraction of fractions.
• Sign Errors with Common Difference: When reversing the AP, remember that the common difference also reverses its sign. A common mistake is to forget this or to use the original common difference.
• Confusing Term Number: Be clear whether you are calculating a term from the beginning or the end. Reversing the sequence is a robust method to avoid confusion.

Summary

The problem requires finding the $20^{th}$ term from the end of a given arithmetic progression. The strategy employed is to recognize that this is equivalent to finding the $20^{th}$ term from the beginning of the reversed arithmetic progression. After converting all terms to improper fractions and calculating the common difference of the original AP, the AP is reversed. The first term of the reversed AP is the last term of the original AP, and its common difference is the negative of the original common difference. Applying the formula for the $n^{th}$ term of an AP to the reversed sequence yields the $20^{th}$ term from the end of the original progression.

The final answer is $\boxed{-115}$.