Question

Write a recursive formula for the explicit formula.
a Subscript nequals5plus​(nminus​1)(negative 4​)
Question content area bottom
Part 1
Choose the correct formula below.
A.
a Subscript nequalsa Subscript n minus 1plus5​; a 1equalsnegative 4
B.
a Subscript n minus 1equalsa Subscript nnegative 4​; a 1equals5
C.
a Subscript n minus 1equalsa Subscript nplus5​; a 1equalsnegative 4
D.
a Subscript nequalsa Subscript n minus 1negative 4​; a 1equals5

Answer 1

To find the correct recursive formula based on the given explicit formula \( a_n = 5 + (n - 1)(-4) \), we can first rewrite and analyze the explicit formula for specific values of \( n \).

The explicit formula simplifies as follows:

\[
a_n = 5 - 4(n - 1)
\]
\[
a_n = 5 - 4n + 4
\]
\[
a_n = 9 - 4n
\]

Now let's compute a few initial values:

- For \( n = 1 \):
\[
a_1 = 9 - 4(1) = 5
\]

- For \( n = 2 \):
\[
a_2 = 9 - 4(2) = 1
\]

- For \( n = 3 \):
\[
a_3 = 9 - 4(3) = -3
\]

- For \( n = 4 \):
\[
a_4 = 9 - 4(4) = -7
\]

- For \( n = 5 \):
\[
a_5 = 9 - 4(5) = -11
\]

Now let's find the differences between consecutive terms:

- \( a_2 - a_1 = 1 - 5 = -4 \)
- \( a_3 - a_2 = -3 - 1 = -4 \)
- \( a_4 - a_3 = -7 - (-3) = -4 \)
- \( a_5 - a_4 = -11 - (-7) = -4 \)

The difference between consecutive terms is constant, equal to -4.

Now we can express the recursive formula:

\[
a_n = a_{n-1} - 4
\]

We also have \( a_1 = 5 \).

Thus, the correct recursive formula corresponds to option D:

**D.** \( a_n = a_{n-1} - 4; \, a_1 = 5 \)