The eighth term of a geometric progression is 128 and common ratio is 2. The product of the first five terms is

Options

A. `4^(6)`
B. `4^(5)`
C. `4^(3)`
D. `4^(8)`

Answer

To solve the problem step by step, we will use the properties of geometric progressions (GP).

### Step 1: Understand the given information
We know:
- The eighth term of the geometric progression (GP) is 128.
- The common ratio (r) is 2.

### Step 2: Write the formula for the nth term of a GP
The nth term of a GP can be expressed as:
\[ a_n = a \cdot r^{n-1} \]
where:
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the term number.

### Step 3: Set up the equation for the eighth term
For the eighth term (n=8):
\[ a_8 = a \cdot r^{8-1} = a \cdot r^7 \]
Substituting the known values:
\[ 128 = a \cdot 2^7 \]

### Step 4: Calculate \( 2^7 \)
Calculate \( 2^7 \):
\[ 2^7 = 128 \]

### Step 5: Substitute and solve for \( a \)
Now substituting back into the equation:
\[ 128 = a \cdot 128 \]
To find \( a \), divide both sides by 128:
\[ a = 1 \]

### Step 6: Write the first five terms of the GP
The first five terms of the GP are:
- First term \( a_1 = a = 1 \)
- Second term \( a_2 = a \cdot r = 1 \cdot 2 = 2 \)
- Third term \( a_3 = a \cdot r^2 = 1 \cdot 2^2 = 4 \)
- Fourth term \( a_4 = a \cdot r^3 = 1 \cdot 2^3 = 8 \)
- Fifth term \( a_5 = a \cdot r^4 = 1 \cdot 2^4 = 16 \)

### Step 7: Calculate the product of the first five terms
The product of the first five terms is:
\[ P = a_1 \cdot a_2 \cdot a_3 \cdot a_4 \cdot a_5 \]
Substituting the values:
\[ P = 1 \cdot 2 \cdot 4 \cdot 8 \cdot 16 \]

### Step 8: Simplify the product
We can simplify this product:
\[ P = 1 \cdot 2 \cdot 4 \cdot 8 \cdot 16 = 2^0 \cdot 2^1 \cdot 2^2 \cdot 2^3 \cdot 2^4 \]
Adding the exponents:
\[ P = 2^{0+1+2+3+4} = 2^{10} \]

### Step 9: Calculate \( 2^{10} \)
Now calculate \( 2^{10} \):
\[ 2^{10} = 1024 \]

### Final Answer
The product of the first five terms is \( 1024 \).

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