Question

Write an explicit formula for the recursive formula.
a Subscript nequalsa Subscript n minus 1plus4.8​; a 1equals9.4
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Part 1
Choose the correct formula.
A.
a Subscript nequalsa Subscript n minus 1plus​(nminus​1)4.8plus9.4
B.
a Subscript nequals9.4plus4.8n
C.
a Subscript nequals9.4plus​(nminus​1)4.8
D.
a Subscript nequals4.8plus​(nminus​1)9.4

Answer 1

To derive the explicit formula from the recursive formula given:

1. The recursive formula is \( a_n = a_{n-1} + 4.8 \) with an initial condition \( a_1 = 9.4 \).

2. Observe that each term is formed by starting from the initial term (\( a_1 \)) and adding \( 4.8 \) for each subsequent term.

3. We can express this relation explicitly:

- \( a_1 = 9.4 \)
- \( a_2 = a_1 + 4.8 = 9.4 + 4.8 \)
- \( a_3 = a_2 + 4.8 = (9.4 + 4.8) + 4.8 = 9.4 + 2 \times 4.8 \)
- \( a_4 = a_3 + 4.8 = (9.4 + 2 \times 4.8) + 4.8 = 9.4 + 3 \times 4.8 \)

Continuing this pattern, we can deduce that:

\[
a_n = 9.4 + (n-1) \cdot 4.8
\]

So, this simplifies to the explicit formula for \( a_n \):

\[
a_n = 9.4 + (n-1) \cdot 4.8
\]

Now let's match this with the provided options:

- A. \( a_n = a_{n-1} + (n-1) \cdot 4.8 + 9.4 \) (not the correct form)
- B. \( a_n = 9.4 + 4.8n \) (not correct, as it has \( n \) instead of \( n-1 \) in the context)
- C. \( a_n = 9.4 + (n-1) \cdot 4.8 \) (this is correct)
- D. \( a_n = 4.8 + (n-1)9.4 \) (not correct)

Thus, the correct answer is **C.** \( a_n = 9.4 + (n-1) \cdot 4.8 \).