Question

find an explicit formula for the geometric sequence 2, 6, 18, 54, .... note: the first term should be b(1). b(n) = \boxed{2} dots 1 / 3 in a geometric sequence, the ratio between successive terms is constant. this means that we can move from any term to the next one by multiplying by a constant value. let’s calculate this ratio over the first few terms: (\frac{54}{18} = \frac{18}{6} = \frac{6}{2} = 3) we see that the constant ratio between successive terms is 3. in other words, we can find any term by starting with the first term and multiplying by 3 repeatedly until we get to the desired term

Explanation:

Step1: Recall the formula for a geometric sequence

The explicit formula for a geometric sequence is given by \( b(n)=b(1)\times r^{n - 1} \), where \( b(1) \) is the first term and \( r \) is the common ratio.

Step2: Identify \( b(1) \) and \( r \)

From the sequence \( 2,6,18,54,\dots \), the first term \( b(1) = 2 \).
To find the common ratio \( r \), we divide a term by its previous term. For example, \( \frac{6}{2}=3 \), \( \frac{18}{6}=3 \), \( \frac{54}{18}=3 \), so \( r = 3 \).

Step3: Substitute \( b(1) \) and \( r \) into the formula

Substituting \( b(1)=2 \) and \( r = 3 \) into \( b(n)=b(1)\times r^{n - 1} \), we get \( b(n)=2\times3^{n - 1} \).

Answer:

\( b(n)=2\times3^{n - 1} \)