Answer:
The Recursive formula for the sequence is:
aₙ = aₙ₋₁ + d
The Explicit formula for the sequence is:
[tex]a_n=4n-1[/tex]
Step-by-step explanation:
Given the sequence
3,7,11,15
Here:
a₁ = 3
computing the differences of all the adjacent terms
7 - 3 = 4, 11 - 7 = 4, 15 - 11 = 4
The difference between all the adjacent terms is the same and equal to
d = 4
We know that a recursive formula basically defines each term of a sequence using the previous term(s).
The recursive formula of the Arithmetic sequence always involves the first term.
a₁ = 3
We know that, in the Arithmetic sequence, every next term can be obtained by adding the common difference and the preceding term.
so
The recursive formula of the sequence is:
aₙ = aₙ₋₁ + d
substitute n = 2 to find the 2nd term
a₂ = a₂₋₁ + d
a₂ = a₁+ d
substitute a₁ = 3 and d = 4
a₂ = 3 + 4
a₂ = 7
Thus, the recursive formula for the sequence 3,7,11,15 is:
aₙ = aₙ₋₁ + d
An explicit rule for the sequence
Given the sequence
3,7,11,15
We already know that
a₁ = 3
d = 4
An arithmetic sequence has a constant difference 'd' and is defined by
[tex]a_n=a_1+\left(n-1\right)d[/tex]
substituting a₁ = 3 and d = 4
[tex]a_n=4\left(n-1\right)+3[/tex]
[tex]a_n=4n-4+3[/tex]
[tex]a_n=4n-1[/tex]
Therefore, an explicit rule for the sequence
[tex]a_n=4n-1[/tex]
