Question

Write an explicit formula for the sequence represented by the following recursive formula.
a Subscript nequalsone third left parenthesis a Subscript n minus 1 Baseline right parenthesis
a 1equals174
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Part 1
a Subscript nequals
  
enter your response here ​(Simplify your​ answer.)
Simplify your​ answer.)

Answer 1

To find an explicit formula for the sequence defined by the recursive relation:

\[
a_n = \frac{1}{3} a_{n-1} \quad \text{with} \quad a_1 = 174,
\]

we recognize that this is a geometric sequence. Specifically, each term is one-third of the previous term.

The first few terms can be computed as follows:

- \( a_1 = 174 \)
- \( a_2 = \frac{1}{3} a_1 = \frac{1}{3} \times 174 = 58 \)
- \( a_3 = \frac{1}{3} a_2 = \frac{1}{3} \times 58 = \frac{58}{3} \)
- \( a_4 = \frac{1}{3} a_3 = \frac{1}{3} \times \frac{58}{3} = \frac{58}{9} \)

From this pattern, we can see that the \( n \)-th term can be expressed in terms of the first term multiplied by the common ratio raised to the power of \( n-1 \):

\[
a_n = a_1 \left(\frac{1}{3}\right)^{n-1}.
\]

Substituting \( a_1 = 174 \):

\[
a_n = 174 \left(\frac{1}{3}\right)^{n-1}.
\]

Thus, the explicit formula for the sequence is:

\[
\boxed{a_n = 174 \cdot \left(\frac{1}{3}\right)^{n-1}}.
\]