Exercise 48

Given the sequence: \( 3, \, 12, \, 48, \, 192, \, \ldots \); If \( n = 0, \, 1, \, 2, \, 3, \ldots \), then an expression describing the sequence is:

  1. \( 3 \cdot 4^{n - 1} \)

  2. \( 3 n \)

  3. \( 3 \cdot 4^n \)

  4. \( 4 n \)

  5. \( 4 \cdot 3^n \)

Try to solve it before checking the answer.
  1. \( 3 \cdot 4^n \)

For \( n = 0, \, 1, \, 2 \)   and   \( 3 \), we have:

\[ 3 \cdot 4^{0} = 3 \cdot 1 = 3, \]
\[ 3 \cdot 4^1 = 12, \]
\[ 3 \cdot 4^2 = 3 \cdot 16 = 48, \]
\[ 3 \cdot 4^3 = 3 \cdot 64 = 192 \]

Hence, an expression that describes, or defines, the secuence is \( \boldsymbol{ 3 \cdot 4^n } \).