Find the 4th term from the end in the progression 3,6,12,…,1536.

Explanation

To find the 4th term from the end in the geometric progression (GP) given by the terms 3, 6, 12, ..., 1536, we can follow these steps:

### Step 1: Identify the first term and common ratio
The first term \( a \) of the GP is 3. The common ratio \( r \) can be calculated by dividing the second term by the first term:
\[ r = \frac{6}{3} = 2 \]

### Step 2: Identify the last term
The last term \( L \) of the GP is given as 1536.

### Step 3: Find the number of terms in the GP
The general formula for the \( n \)-th term of a geometric progression is given by:
\[ a_n = a \cdot r^{n-1} \]
Setting \( a_n = 1536 \), we can solve for \( n \):
\[ 1536 = 3 \cdot 2^{n-1} \]
Dividing both sides by 3:
\[ 512 = 2^{n-1} \]
Since \( 512 = 2^9 \), we can equate the exponents:
\[ n - 1 = 9 \implies n = 10 \]
Thus, there are 10 terms in the GP.

### Step 4: Find the 4th term from the end
To find the 4th term from the end, we can use the formula for the \( n \)-th term from the end:
\[ a_{n-k+1} = \frac{L}{r^{k-1}} \]
where \( L \) is the last term, \( r \) is the common ratio, and \( k \) is the position from the end. Here, \( k = 4 \):
\[ a_{10-4+1} = \frac{1536}{2^{4-1}} = \frac{1536}{2^3} = \frac{1536}{8} \]
Calculating this gives:
\[ \frac{1536}{8} = 192 \]

### Conclusion
The 4th term from the end in the given geometric progression is **192**.

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