How many terms of the series `1+3+9+` .. .........sum to 364?

Options

A. 5
B. 6
C. 8
D. 9

Answer

To solve the problem of how many terms of the series \(1 + 3 + 9 + \ldots\) sum to 364, we can follow these steps:

### Step 1: Identify the Type of Series
The given series is \(1, 3, 9, \ldots\). This is a geometric progression (GP) where:
- The first term \(a = 1\)
- The common ratio \(r = \frac{3}{1} = 3\)

### Step 2: Write the Formula for the Sum of the First \(n\) Terms of a GP
The formula for the sum of the first \(n\) terms \(S_n\) of a geometric series is given by:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
Substituting the values of \(a\) and \(r\):
\[ S_n = 1 \cdot \frac{1 - 3^n}{1 - 3} = \frac{1 - 3^n}{-2} = \frac{3^n - 1}{2} \]

### Step 3: Set Up the Equation
We are given that the sum \(S_n = 364\). Therefore, we can set up the equation:
\[ \frac{3^n - 1}{2} = 364 \]

### Step 4: Solve for \(3^n\)
Multiply both sides by 2 to eliminate the fraction:
\[ 3^n - 1 = 728 \]
Now, add 1 to both sides:
\[ 3^n = 729 \]

### Step 5: Express 729 as a Power of 3
We know that:
\[ 729 = 3^6 \]
Thus, we can equate the powers:
\[ 3^n = 3^6 \]

### Step 6: Solve for \(n\)
By comparing the exponents, we find:
\[ n = 6 \]

### Conclusion
The number of terms in the series that sum to 364 is \(n = 6\).

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