A recursive rule for an arithmetic sequence is a1 = 4; an = an-1 + 3 . What is an explicit rule for this sequence? (Picture attached) please help? I don't remember what type of equation to use for these kinds of problems...

Answer: 3n+1
(no need to type in the "a_n" or "an" part, as it's already done)

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Explanation:

The first term is a1 = 4 since it is given to us
The common difference is d = 3. This is the value that we add to each term to get the next as the recursive steps shows when we write a(n) = a(n-1)+3
In other words,
nth term = [ (n-1)st term ] + 3
next term = (previous term) + 3

Using a1 = 4 and d = 3, we get the following
an = a1+d(n-1)
an = 4 + 3(n-1)
an = 4+3n-3
an = 3n + (4-3)
an = 3n+1

ibiscollingwood ibiscollingwood · Answer 2 · 2021-11-25T16:05:16+01:00

An arithmetic sequence is a sequence where each term increases by adding/subtracting some constant d and d is known as common difference.

General term of given sequence is, aₙ = 3n + 1

Given data are, a₁ = 4 , [tex]a_{n}=a_{n-1}+3[/tex]

[tex]a_{n}=a_{n-1}+3\\\\a_{n}-a_{n-1}=3\\[/tex]

[tex]a_{n}-a_{n-1}=d[/tex] is known as common difference.

So, [tex]d=a_{n}-a_{n-1}=3[/tex]

General term for an arithmetic sequence is,

[tex]a_{n}=a_{1}+(n-1)d\\\\a_{n}=4+(n-1)3\\\\a_{n}=4+3n-3\\\\a_{n}=3n+1[/tex]

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