(a) Find the value of 'x if x + 1, 2x + 1 and x + 1 are in A.P. Also find the 4th term of this progression.
(b) If k + 3, 2k + 1, k + 7 are in A.P., then find this progression upto 5 terms.

Explanation

### Solution:

**Part (a):**

1. **Identify the terms in A.P.**
We have the terms:
\( a_1 = x + 1 \)
\( a_2 = 2x + 1 \)
\( a_3 = x + 7 \)

2. **Use the property of A.P.**
For three terms to be in A.P., the middle term must be equal to the average of the other two terms:
\[ 2a_2 = a_1 + a_3 \]
Substituting the values:
\[ 2(2x + 1) = (x + 1) + (x + 7) \]

3. **Simplify the equation:**
\[ 4x + 2 = x + 1 + x + 7 \]
\[ 4x + 2 = 2x + 8 \]

4. **Rearranging the equation:**
\[ 4x - 2x = 8 - 2 \]
\[ 2x = 6 \]
\[ x = 3 \]

5. **Find the terms of the A.P. using \( x = 3 \):**
\[ a_1 = 3 + 1 = 4 \]
\[ a_2 = 2(3) + 1 = 7 \]
\[ a_3 = 3 + 7 = 10 \]

6. **Find the common difference \( d \):**
\[ d = a_2 - a_1 = 7 - 4 = 3 \]

7. **Find the 4th term \( a_4 \):**
\[ a_4 = a_1 + 3d = 4 + 3(3) = 4 + 9 = 13 \]

**Final Answer for Part (a):**
The value of \( x \) is \( 3 \) and the 4th term of the A.P. is \( 13 \).

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**Part (b):**

1. **Identify the terms in A.P.:**
We have the terms:
\( a_1 = k + 3 \)
\( a_2 = 2k + 1 \)
\( a_3 = k + 7 \)

2. **Use the property of A.P.:**
\[ 2a_2 = a_1 + a_3 \]
Substituting the values:
\[ 2(2k + 1) = (k + 3) + (k + 7) \]

3. **Simplify the equation:**
\[ 4k + 2 = k + 3 + k + 7 \]
\[ 4k + 2 = 2k + 10 \]

4. **Rearranging the equation:**
\[ 4k - 2k = 10 - 2 \]
\[ 2k = 8 \]
\[ k = 4 \]

5. **Find the terms of the A.P. using \( k = 4 \):**
\[ a_1 = 4 + 3 = 7 \]
\[ a_2 = 2(4) + 1 = 9 \]
\[ a_3 = 4 + 7 = 11 \]

6. **Find the common difference \( d \):**
\[ d = a_2 - a_1 = 9 - 7 = 2 \]

7. **Find the 4th term \( a_4 \):**
\[ a_4 = a_1 + 3d = 7 + 3(2) = 7 + 6 = 13 \]

8. **Find the 5th term \( a_5 \):**
\[ a_5 = a_1 + 4d = 7 + 4(2) = 7 + 8 = 15 \]

**Final Answer for Part (b):**
The progression up to 5 terms is \( 7, 9, 11, 13, 15 \).

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