If 20th term from the end of the progression `20, "19"1/4, "18"1/2,"17"3/4......,"-129"1/4` is

Options

A. -120
B. -115
C. -125
D. -110

Answer

To find the 20th term from the end of the arithmetic progression given by the sequence \(20, 19\frac{1}{4}, 18\frac{1}{2}, 17\frac{3}{4}, \ldots, -129\frac{1}{4}\), we can follow these steps:

### Step 1: Identify the first term and the last term
The first term \(a\) of the sequence is:
\[ a = 20 \]
The last term \(l\) of the sequence is:
\[ l = -129\frac{1}{4} = -\frac{517}{4} \]

### Step 2: Determine the common difference \(d\)
To find the common difference \(d\), we can subtract the second term from the first term:
\[ d = 19\frac{1}{4} - 20 = 19.25 - 20 = -0.75 = -\frac{3}{4} \]

### Step 3: Calculate the total number of terms \(n\)
To find the total number of terms \(n\) in the sequence, we can use the formula for the \(n\)th term of an arithmetic progression:
\[ l = a + (n-1)d \]
Substituting the known values:
\[ -\frac{517}{4} = 20 + (n-1)\left(-\frac{3}{4}\right) \]
Rearranging gives:
\[ -\frac{517}{4} - 20 = (n-1)\left(-\frac{3}{4}\right) \]
Converting 20 to a fraction:
\[ 20 = \frac{80}{4} \]
Thus:
\[ -\frac{517}{4} - \frac{80}{4} = (n-1)\left(-\frac{3}{4}\right) \]
\[ -\frac{597}{4} = (n-1)\left(-\frac{3}{4}\right) \]
Multiplying both sides by \(-\frac{4}{3}\):
\[ n - 1 = \frac{597}{3} = 199 \]
So:
\[ n = 200 \]

### Step 4: Find the 20th term from the end
The 20th term from the end can be calculated using the formula for the \(k\)th term from the end:
\[ T_k = l + (k-1)d \]
Substituting \(k = 20\):
\[ T_{20} = -\frac{517}{4} + (20-1)\left(-\frac{3}{4}\right) \]
\[ T_{20} = -\frac{517}{4} + 19\left(-\frac{3}{4}\right) \]
Calculating:
\[ T_{20} = -\frac{517}{4} - \frac{57}{4} = -\frac{574}{4} = -143.5 \]

### Final Answer
Thus, the 20th term from the end of the progression is:
\[ \boxed{-143.5} \]