If sum of n terms of a sequence is given by `S _(n) = 2n ^(2) - 4n,` find its `50^(th)` term.

Answer

To find the 50th term of the sequence where the sum of the first n terms is given by \( S_n = 2n^2 - 4n \), we can follow these steps:

### Step 1: Understand the relationship between the sum of terms and individual terms
The nth term of the sequence, denoted as \( T_n \), can be found using the formula:
\[ T_n = S_n - S_{n-1} \]
This means that to find the nth term, we need to subtract the sum of the first \( n-1 \) terms from the sum of the first \( n \) terms.

### Step 2: Calculate \( S_{50} \)
Using the given formula for \( S_n \):
\[ S_{50} = 2(50^2) - 4(50) \]
Calculating this step by step:
\[ 50^2 = 2500 \]
\[ S_{50} = 2(2500) - 4(50) = 5000 - 200 = 4800 \]

### Step 3: Calculate \( S_{49} \)
Now we need to calculate \( S_{49} \):
\[ S_{49} = 2(49^2) - 4(49) \]
Calculating this step by step:
\[ 49^2 = 2401 \]
\[ S_{49} = 2(2401) - 4(49) = 4802 - 196 = 4606 \]

### Step 4: Find \( T_{50} \)
Now, we can find \( T_{50} \):
\[ T_{50} = S_{50} - S_{49} = 4800 - 4606 \]
Calculating this gives:
\[ T_{50} = 194 \]

### Final Answer
Thus, the 50th term of the sequence is:
\[ \boxed{194} \]