The recursive rule for a sequence is shown. an=an−1+7 a1=17 what is the explicit rule for this sequence?

Given the recursive formula:
an=a(n-1)+7; a1=17
to get the explicit formula we proceed as follows:
common difference=7
a1=17
but the explicit formula for arithmetic sequence is:
an=a+(n-1)d
where:
a=fist term
n=number of terms
d=common difference
plugging the values in the formula we get:
an=17+(n-1)7
simplifying we get
an=17+7n-7
an=7n+10

Answer: an=7n+10

Answer Link

MrRoyal MrRoyal · Answer 2 · 2022-03-22T12:36:07+01:00

The explicit rule for the sequence is [tex]a_n = 10 + 7n[/tex]

The recursive rule is given as:

[tex]a_n=a_{n-1}+7[/tex]

[tex]a_1 = 17[/tex]

The above shows that the sequence is an arithmetic sequence.

Start by calculating a2

We have:

[tex]a_n=a_{n-1}+7[/tex]

Substitute 2 for n

[tex]a_2=a_1+7[/tex]

This gives

[tex]a_2 = 17 + 7[/tex]

[tex]a_2 = 24[/tex]

So, we have:

a1=17

a2 = 24

The common difference (d) is:

d =a2 - a1

d = 24 - 17

d = 7

The explicit rule is then calculated as:

[tex]a_n = a_1 + (n - 1)d[/tex]

This gives

[tex]a_n = 17 + (n - 1) * 7[/tex]

Expand

[tex]a_n = 17 - 7 + 7n[/tex]

[tex]a_n = 10 + 7n[/tex]

Hence, the explicit rule for the sequence is [tex]a_n = 10 + 7n[/tex]

Read more about arithmetic sequence at:

https://brainly.com/question/6561461