Find the 6th and nth terms of the GP 2, 6, 18, 54, ...

Answer

To solve the problem of finding the 6th and nth terms of the given geometric progression (GP) 2, 6, 18, 54, we can follow these steps:

### Step 1: Identify the first term (a) and the common ratio (r)
The first term \( a \) of the GP is the first number in the sequence, which is:
\[ a = 2 \]

Next, we calculate the common ratio \( r \) by dividing the second term by the first term:
\[ r = \frac{6}{2} = 3 \]

### Step 2: Find the 6th term of the GP
The formula for the \( n \)th term of a GP is given by:
\[ T_n = a \cdot r^{n-1} \]

To find the 6th term, we substitute \( n = 6 \):
\[ T_6 = a \cdot r^{6-1} = 2 \cdot 3^{5} \]

Now we calculate \( 3^5 \):
\[ 3^5 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 243 \]

Now substitute this value back into the equation for \( T_6 \):
\[ T_6 = 2 \cdot 243 = 486 \]

### Step 3: Find the nth term of the GP
Using the same formula for the nth term, we have:
\[ T_n = a \cdot r^{n-1} \]
Substituting the known values:
\[ T_n = 2 \cdot 3^{n-1} \]

### Final Answers
- The 6th term of the GP is \( 486 \).
- The nth term of the GP is \( 2 \cdot 3^{n-1} \).

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