The second term of a G.P. is 2 and the sum of infinite terms is 8. Find the first term.

Answer

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

### Step 1: Understand the given information
We know that:
- The second term of the G.P. (Geometric Progression) is 2.
- The sum of the infinite terms of the G.P. is 8.

### Step 2: Define the terms
Let:
- \( A \) be the first term of the G.P.
- \( r \) be the common ratio of the G.P.

The second term of the G.P. can be expressed as:
\[ A_2 = A \cdot r = 2 \]

### Step 3: Use the formula for the sum of an infinite G.P.
The formula for the sum of an infinite G.P. is given by:
\[ S = \frac{A}{1 - r} \]
According to the problem, the sum \( S \) is 8, so we can write:
\[ 8 = \frac{A}{1 - r} \]

### Step 4: Express \( r \) in terms of \( A \)
From the second term equation \( A \cdot r = 2 \), we can express \( r \) as:
\[ r = \frac{2}{A} \]

### Step 5: Substitute \( r \) in the sum formula
Now, substitute \( r \) in the sum formula:
\[ 8 = \frac{A}{1 - \frac{2}{A}} \]

### Step 6: Simplify the equation
First, simplify the denominator:
\[ 1 - \frac{2}{A} = \frac{A - 2}{A} \]
So, the equation becomes:
\[ 8 = \frac{A}{\frac{A - 2}{A}} = \frac{A^2}{A - 2} \]

### Step 7: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 8(A - 2) = A^2 \]
Expanding this, we have:
\[ 8A - 16 = A^2 \]

### Step 8: Rearrange the equation
Rearranging the equation leads to:
\[ A^2 - 8A + 16 = 0 \]

### Step 9: Factor the quadratic equation
This can be factored as:
\[ (A - 4)^2 = 0 \]

### Step 10: Solve for \( A \)
Setting the factor equal to zero gives:
\[ A - 4 = 0 \implies A = 4 \]

### Final Answer
The first term \( A \) of the G.P. is \( 4 \).

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