How many terms of the series 1+2+4+…. Has the sum 511 ?

Options

A. 7
B. 8
C. 9
D. 10

Explanation

To find how many terms of the series \(1 + 2 + 4 + \ldots\) have a sum of \(511\), we can follow these steps:

### Step 1: Identify the series type
The series given is a geometric progression (GP) where:
- The first term \(a = 1\)
- The common ratio \(r = 2\)

### 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 of a GP is given by:
\[ S_n = \frac{a(r^n - 1)}{r - 1} \]
Substituting the values of \(a\) and \(r\):
\[ S_n = \frac{1(2^n - 1)}{2 - 1} = 2^n - 1 \]

### Step 3: Set the sum equal to 511
We know from the problem that the sum \(S_n = 511\). Therefore, we can set up the equation:
\[ 2^n - 1 = 511 \]

### Step 4: Solve for \(n\)
Adding \(1\) to both sides gives:
\[ 2^n = 512 \]

### Step 5: Express \(512\) as a power of \(2\)
We can express \(512\) as a power of \(2\):
\[ 512 = 2^9 \]
Thus, we have:
\[ 2^n = 2^9 \]

### Step 6: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ n = 9 \]

### Conclusion
The number of terms \(n\) in the series that sum to \(511\) is \(9\).