An atom has x energy level , then total number of lines in its spectrum are :

Options

A. `1 + 2+ 3 …… (x +1)`
B. `1 + 2 + 3 ….. (x^(2))`
C. `1 + 2 +3 …… (x -1)`
D. `(x + 1) (x+2) (x+4)`

Answer

To determine the total number of lines in the spectrum of an atom with \( x \) energy levels, we can follow these steps:

### Step 1: Understand the Concept of Energy Levels
An atom can exist in various energy levels, denoted by \( n \), where \( n \) can take values from 1 to \( x \). The transitions between these energy levels result in the emission or absorption of photons, which correspond to spectral lines.

### Step 2: Determine Possible Transitions
When an electron transitions from a higher energy level to a lower energy level, it emits a photon, creating a spectral line. The number of possible transitions can be calculated using the formula for combinations.

### Step 3: Calculate the Total Number of Spectral Lines
The total number of spectral lines can be found using the formula:
\[ \text{Total Lines} = \frac{n(n-1)}{2} \]
Where \( n \) is the number of energy levels. In our case, \( n = x \).

### Step 4: Substitute \( x \) into the Formula
Substituting \( x \) into the formula gives:
\[ \text{Total Lines} = \frac{x(x-1)}{2} \]

### Step 5: Conclusion
Thus, the total number of lines in the spectrum of an atom with \( x \) energy levels is:
\[ \frac{x(x-1)}{2} \]