If all the terms of a geometric sseries are positive, the first term of the series is 2, and the third term is 8, how many digits are there in the 40th term ?

Options

A. 10
B. 11
C. 12
D. 13

Explanation

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Identify the first term and the third term
We are given that the first term \( a = 2 \) and the third term \( T_3 = 8 \).

### Step 2: Use the formula for the nth term of a geometric progression (GP)
The formula for the nth term of a geometric progression is given by:
\[ T_n = a \cdot r^{n-1} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

### Step 3: Set up the equation for the third term
For the third term:
\[ T_3 = a \cdot r^{3-1} = a \cdot r^2 \]
Substituting the known values:
\[ 8 = 2 \cdot r^2 \]

### Step 4: Solve for the common ratio \( r \)
To find \( r^2 \), we rearrange the equation:
\[ r^2 = \frac{8}{2} = 4 \]
Taking the square root of both sides, we get:
\[ r = \sqrt{4} = 2 \]
Since all terms are positive, we take the positive root.

### Step 5: Find the 40th term
Now, we need to find the 40th term \( T_{40} \):
\[ T_{40} = a \cdot r^{40-1} = 2 \cdot r^{39} \]
Substituting \( r = 2 \):
\[ T_{40} = 2 \cdot 2^{39} = 2^{1 + 39} = 2^{40} \]

### Step 6: Calculate \( 2^{40} \)
We need to find the value of \( 2^{40} \). Using a calculator or known values:
\[ 2^{40} \approx 1.0995 \times 10^{12} \]

### Step 7: Determine the number of digits in \( 2^{40} \)
The number of digits \( d \) in a number \( x \) can be found using the formula:
\[ d = \lfloor \log_{10} x \rfloor + 1 \]
For \( x = 2^{40} \):
\[ \log_{10}(2^{40}) = 40 \cdot \log_{10}(2) \approx 40 \cdot 0.3010 \approx 12.04 \]
Thus,
\[ d = \lfloor 12.04 \rfloor + 1 = 12 + 1 = 13 \]

### Final Answer
The number of digits in the 40th term \( T_{40} \) is **13**.

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