Find the number of terms in the G.P. 1, 2, 4, 8, ... 4096.

Explanation

To find the number of terms in the geometric progression (G.P.) given by 1, 2, 4, 8, ..., 4096, we can follow these steps:

### Step 1: Identify the first term and common ratio
The first term \( a \) of the G.P. is 1. The common ratio \( r \) can be calculated by dividing the second term by the first term:
\[ r = \frac{2}{1} = 2 \]

**Hint:** The first term is the initial value of the sequence, and the common ratio is found by dividing any term by its preceding term.

### Step 2: Write the formula for the nth term of a G.P.
The formula for the nth term of a G.P. is given by:
\[ a_n = a \cdot r^{n-1} \]
where \( a_n \) is the nth term, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

**Hint:** This formula helps us express any term in the sequence based on its position.

### Step 3: Set up the equation for the last term
In this case, we know the last term \( a_n \) is 4096. Substituting the known values into the formula gives:
\[ 4096 = 1 \cdot 2^{n-1} \]
This simplifies to:
\[ 4096 = 2^{n-1} \]

**Hint:** Substitute the known last term into the nth term formula to find \( n \).

### Step 4: Express 4096 as a power of 2
We can express 4096 as a power of 2:
\[ 4096 = 2^{12} \]

**Hint:** Recognizing numbers as powers of their base can help simplify the equation.

### Step 5: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:
\[ n - 1 = 12 \]

**Hint:** When bases are the same, their exponents must also be equal.

### Step 6: Solve for \( n \)
Now, solve for \( n \):
\[ n = 12 + 1 = 13 \]

**Hint:** Adding 1 accounts for the first term in the sequence.

### Conclusion
The number of terms in the G.P. is 13.

**Final Answer:** 13