The first, eighth and twenty-second terms of an A.P. are three consecutive terms of a G.P. Find the common ratio of the G.P. Given also that the sum of the first twenty-two terms of the A.P.is 275, find its first term.

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Identify the terms of the A.P.
Let the first term of the A.P. be \( A \) and the common difference be \( D \). The terms we are interested in are:
- First term \( A_1 = A \)
- Eighth term \( A_8 = A + 7D \)
- Twenty-second term \( A_{22} = A + 21D \)

### Step 2: Set up the relationship for the G.P.
Since \( A_1, A_8, A_{22} \) are three consecutive terms of a G.P., we can use the property of G.P. that states the square of the middle term is equal to the product of the other two terms:
\[ (A + 7D)^2 = A \cdot (A + 21D) \]

### Step 3: Expand both sides of the equation.
Expanding the left side:
\[ (A + 7D)^2 = A^2 + 14AD + 49D^2 \]
Expanding the right side:
\[ A \cdot (A + 21D) = A^2 + 21AD \]

### Step 4: Set the two expansions equal to each other.
Now we have:
\[ A^2 + 14AD + 49D^2 = A^2 + 21AD \]

### Step 5: Simplify the equation.
Subtract \( A^2 \) from both sides:
\[ 14AD + 49D^2 = 21AD \]
Rearranging gives:
\[ 49D^2 = 21AD - 14AD \]
\[ 49D^2 = 7AD \]

### Step 6: Solve for \( A \) in terms of \( D \).
Dividing both sides by \( D \) (assuming \( D \neq 0 \)):
\[ 49D = 7A \]
Thus,
\[ A = 7D \]

### Step 7: Use the sum of the first 22 terms of the A.P.
The sum of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \left(2A + (n - 1)D\right) \]
For \( n = 22 \):
\[ S_{22} = \frac{22}{2} \left(2A + 21D\right) = 11(2A + 21D) \]
We know \( S_{22} = 275 \):
\[ 11(2A + 21D) = 275 \]
Dividing both sides by 11:
\[ 2A + 21D = 25 \]

### Step 8: Substitute \( A = 7D \) into the equation.
Substituting \( A \):
\[ 2(7D) + 21D = 25 \]
This simplifies to:
\[ 14D + 21D = 25 \]
\[ 35D = 25 \]
Thus,
\[ D = \frac{25}{35} = \frac{5}{7} \]

### Step 9: Find the first term \( A \).
Substituting \( D \) back into \( A = 7D \):
\[ A = 7 \cdot \frac{5}{7} = 5 \]

### Step 10: Find the common ratio \( R \) of the G.P.
The common ratio \( R \) is given by:
\[ R = \frac{A_8}{A} = \frac{A + 7D}{A} \]
Substituting \( A \) and \( D \):
\[ R = \frac{5 + 7 \cdot \frac{5}{7}}{5} = \frac{5 + 5}{5} = \frac{10}{5} = 2 \]

### Final Answers:
- The common ratio \( R \) of the G.P. is \( 2 \).
- The first term \( A \) of the A.P. is \( 5 \).