The sum to infinity of the progression `9-3 + 1-1/3 + `……..is

Options

A. A) 9
B. B) `9//2`
C. C) `27//4`
D. D) `15//2`

Answer

To find the sum to infinity of the given progression \(9 - 3 + 1 - \frac{1}{3} + \ldots\), we can follow these steps:

### Step 1: Identify the first term (A) and the common ratio (R)
The first term of the series is clearly \(A = 9\).

Next, we need to determine the common ratio \(R\). We can find the common ratio by dividing the second term by the first term:
\[ R = \frac{-3}{9} = -\frac{1}{3} \]

### Step 2: Confirm the common ratio with subsequent terms
To ensure that the ratio is consistent, we can check the ratio of the next pair of terms:
\[ R = \frac{1}{-3} = -\frac{1}{3} \]
This confirms that the common ratio is indeed \(R = -\frac{1}{3}\).

### Step 3: Use the formula for the sum to infinity of a geometric progression
The formula for the sum to infinity \(S_{\infty}\) of a geometric progression is given by:
\[ S_{\infty} = \frac{A}{1 - R} \]
where \(A\) is the first term and \(R\) is the common ratio.

### Step 4: Substitute the values into the formula
Substituting \(A = 9\) and \(R = -\frac{1}{3}\) into the formula:
\[ S_{\infty} = \frac{9}{1 - (-\frac{1}{3})} \]
This simplifies to:
\[ S_{\infty} = \frac{9}{1 + \frac{1}{3}} = \frac{9}{\frac{4}{3}} \]

### Step 5: Simplify the expression
To simplify \(\frac{9}{\frac{4}{3}}\), we multiply by the reciprocal:
\[ S_{\infty} = 9 \times \frac{3}{4} = \frac{27}{4} \]

### Final Answer
Thus, the sum to infinity of the progression is:
\[ \boxed{\frac{27}{4}} \]

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