In the following A.P. 's find the missing terms in the boxes :
`square, 38, square,square,square,-22`

Answer

To find the missing terms in the arithmetic progression (A.P.) represented as `square, 38, square, square, square, -22`, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Terms**:
Let the terms be represented as follows:
- \( a_1 = \text{square} \)
- \( a_2 = 38 \)
- \( a_3 = \text{square} \)
- \( a_4 = \text{square} \)
- \( a_5 = \text{square} \)
- \( a_6 = -22 \)

2. **Use the A.P. Formula**:
The general formula for the nth term of an A.P. is:
\[ a_n = a_1 + (n-1)d \]
where \( d \) is the common difference.

3. **Express \( a_2 \) and \( a_6 \)**:
From the formula, we can express \( a_2 \) and \( a_6 \):
- \( a_2 = a_1 + d \) (for \( n=2 \))
- \( a_6 = a_1 + 5d \) (for \( n=6 \))

4. **Set Up the Equations**:
We know:
\[ 38 = a_1 + d \quad \text{(1)} \]
\[ -22 = a_1 + 5d \quad \text{(2)} \]

5. **Subtract Equation (1) from Equation (2)**:
To eliminate \( a_1 \), we can subtract equation (1) from equation (2):
\[ (-22) - 38 = (a_1 + 5d) - (a_1 + d) \]
Simplifying gives:
\[ -60 = 4d \]
Therefore,
\[ d = -15 \]

6. **Substitute \( d \) back into Equation (1)**:
Now substitute \( d \) back into equation (1) to find \( a_1 \):
\[ 38 = a_1 - 15 \]
So,
\[ a_1 = 38 + 15 = 53 \]

7. **Calculate the Missing Terms**:
Now we can find the missing terms:
- \( a_3 = a_1 + 2d = 53 + 2(-15) = 53 - 30 = 23 \)
- \( a_4 = a_1 + 3d = 53 + 3(-15) = 53 - 45 = 8 \)
- \( a_5 = a_1 + 4d = 53 + 4(-15) = 53 - 60 = -7 \)

### Final Values:
- \( a_1 = 53 \)
- \( a_2 = 38 \)
- \( a_3 = 23 \)
- \( a_4 = 8 \)
- \( a_5 = -7 \)
- \( a_6 = -22 \)

Thus, the missing terms in the boxes are:
- First box: 53
- Third box: 23
- Fourth box: 8
- Fifth box: -7