5. A sequence is defined by $$ f(1)=3 $$ and $$ f(n)=2 \cdot f(n-1) $$ for $$ n \geq 2 $$. Which of the $$ \quad $$ * 1 point following statements defines the $$ n $$-th term of $$ f $$ ?

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To find the $$ n $$-th term of the sequence defined by $$ f(1) = 3 $$ and $$ f(n) = 2 \cdot f(n-1) $$ for $$ n \geq 2 $$, we can recognize that this is a geometric sequence where each term is twice the previous term.

The general formula for the $$ n $$-th term of a geometric sequence is:
$$
f(n) = f(1) \cdot r^{n-1}
$$
where $$ r $$ is the common ratio. In this case, $$ r = 2 $$ and $$ f(1) = 3 $$.

Plugging in these values:
$$
f(n) = 3 \cdot 2^{n-1}
$$

**Answer:**  
For all $$ n \geq 1 $$, the $$ n $$-th term is $$ f(n) = 3 \times 2^{\,n-1} $$.
The $$ n $$-th term of the sequence is $$ f(n) = 3 \times 2^{n-1} $$.

5. A sequence is defined by \( f(1)=3 \) and \( f(n)=2 \cdot f(n-1) \) for \( n \geq 2 \). Which of the \( \quad \) * 1 point following statements defines the \( n \)-th term of \( f \) ?

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