Find the seventh term from the end of the series : `sqrt(2),2,2sqrt(2), . . . .. . . . ,32`.

Answer

To find the seventh term from the end of the geometric progression (GP) series: \( \sqrt{2}, 2, 2\sqrt{2}, \ldots, 32 \), we can follow these steps:

### Step 1: Identify the first term and common ratio
The first term \( a \) of the series is \( \sqrt{2} \).

To find the common ratio \( r \):
- The second term is \( 2 \).
- Thus, \( r = \frac{2}{\sqrt{2}} = \sqrt{2} \).

### Step 2: Find the last term
The last term given in the series is \( 32 \).

### Step 3: Use the formula for the nth term of a GP
The formula for the nth term of a GP is given by:
\[ T_n = a \cdot r^{n-1} \]
Setting \( T_n = 32 \):
\[ 32 = \sqrt{2} \cdot (\sqrt{2})^{n-1} \]

### Step 4: Simplify the equation
This can be rewritten as:
\[ 32 = \sqrt{2} \cdot 2^{(n-1)/2} \]
Since \( \sqrt{2} = 2^{1/2} \), we can express the equation as:
\[ 32 = 2^{1/2} \cdot 2^{(n-1)/2} \]
Combining the exponents:
\[ 32 = 2^{(1 + n - 1)/2} = 2^{n/2} \]

### Step 5: Convert 32 to a power of 2
We know that \( 32 = 2^5 \), so we can set the exponents equal:
\[ \frac{n}{2} = 5 \]
Multiplying both sides by 2 gives:
\[ n = 10 \]

### Step 6: Find the total number of terms
The total number of terms in the GP is \( n = 10 \).

### Step 7: Find the seventh term from the end
To find the seventh term from the end, we can use the formula for the \( k \)-th term from the end:
\[ T_{n-k+1} = T_{10-7+1} = T_4 \]

### Step 8: Calculate \( T_4 \)
Using the nth term formula again:
\[ T_4 = a \cdot r^{4-1} = \sqrt{2} \cdot (\sqrt{2})^{3} = \sqrt{2} \cdot 2^{3/2} = \sqrt{2} \cdot 2 \cdot \sqrt{2} = 2 \cdot 2 = 4 \]

### Final Answer
Thus, the seventh term from the end of the series is \( 4 \).

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