The middle term of the progression `20,16,12, cdots , -176,-180` is (a)-46 (b)-76 (c)-80 (d) None of these

Options

A. -46
B. -76
C. -80
D. none of these

Answer

To find the middle term of the arithmetic progression (AP) given by the sequence \(20, 16, 12, \ldots, -176, -180\), we will follow these steps:

### Step 1: Identify the first term and common difference
The first term \(a\) of the sequence is:
\[ a = 20 \]
The common difference \(d\) can be calculated as:
\[ d = 16 - 20 = -4 \]

### Step 2: Determine the last term
The last term of the sequence is given as \(-180\).

### Step 3: Use the formula for the nth term of an AP
The nth term of an AP can be expressed as:
\[ T_n = a + (n-1)d \]
Setting \(T_n = -180\), we can write:
\[ -180 = 20 + (n-1)(-4) \]

### Step 4: Solve for \(n\)
Rearranging the equation:
\[ -180 - 20 = (n-1)(-4) \]
\[ -200 = (n-1)(-4) \]
Dividing both sides by \(-4\):
\[ n-1 = \frac{-200}{-4} = 50 \]
Thus:
\[ n = 50 + 1 = 51 \]

### Step 5: Find the middle term
Since there are 51 terms, the middle term is the \(\frac{n+1}{2}\)th term:
\[ \text{Middle term position} = \frac{51 + 1}{2} = 26 \]

### Step 6: Calculate the 26th term
Now we need to find the 26th term \(T_{26}\):
\[ T_{26} = a + (26-1)d = 20 + 25(-4) \]
Calculating this gives:
\[ T_{26} = 20 - 100 = -80 \]

### Conclusion
The middle term of the progression is:
\[ \boxed{-80} \]