Out of 2n terms , n terms are 'a' and rest are '-a'. If we add 'b' to all the terms than mean is 5 and standard deviation is 20 then `a^2+b^2=?`

Answer

To solve the problem, we need to find the value of \( a^2 + b^2 \) given that out of \( 2n \) terms, \( n \) terms are \( a \) and the rest are \( -a \). We also know that after adding \( b \) to all terms, the mean is 5 and the standard deviation is 20.

### Step-by-step Solution:

1. **Understanding the Terms**:
We have \( n \) terms equal to \( a \) and \( n \) terms equal to \( -a \). Therefore, the total number of terms is \( 2n \).

2. **Calculating the Old Mean**:
The sum of the terms is:
\[ \text{Sum} = n \cdot a + n \cdot (-a) = 0 \]
Thus, the old mean is:
\[ \text{Old Mean} = \frac{\text{Sum}}{2n} = \frac{0}{2n} = 0 \]

3. **Calculating the New Mean**:
When we add \( b \) to each term, the new mean becomes:
\[ \text{New Mean} = \frac{\text{Sum} + 2nb}{2n} = \frac{0 + 2nb}{2n} = b \]
We are given that the new mean is 5, hence:
\[ b = 5 \]

4. **Calculating the Standard Deviation**:
The standard deviation is defined as:
\[ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \]
Since we added \( b \) to each term, the mean remains unchanged in terms of standard deviation calculation. Therefore, we can calculate the standard deviation based on the original terms.

The variance (which is the square of the standard deviation) can be calculated as follows:
\[ \sigma^2 = \frac{\sum (x_i^2)}{2n} - \left(\frac{\sum x_i}{2n}\right)^2 \]
Since the old mean is 0, the second term becomes 0:
\[ \sigma^2 = \frac{\sum (x_i^2)}{2n} \]
The terms \( x_i \) are \( a \) and \( -a \). Therefore:
\[ \sum (x_i^2) = n \cdot a^2 + n \cdot (-a)^2 = n \cdot a^2 + n \cdot a^2 = 2n \cdot a^2 \]
Thus, we have:
\[ \sigma^2 = \frac{2n \cdot a^2}{2n} = a^2 \]
We are given that the standard deviation is 20, hence:
\[ a^2 = 20^2 = 400 \]

5. **Calculating \( a^2 + b^2 \)**:
Now we have:
\[ a^2 = 400 \quad \text{and} \quad b = 5 \implies b^2 = 5^2 = 25 \]
Therefore:
\[ a^2 + b^2 = 400 + 25 = 425 \]

### Final Answer:
\[ a^2 + b^2 = 425 \]