The choice which defines same arithmatic sequence using a Recursive Formula and an Explicit formula is (A) which is [tex]a_1= -23, a_n = a_{n-1}+ 12,a_n=-23 +12(n-1)[/tex]
If first term of a arithmatic sequence is a₁ and the common difference is d then,
By Explicit Formula, the n-th term of the sequence is given by,
[tex]a_n=a_1+(n-1)d[/tex]
By Recursive Formula,
[tex]a_n=a_{n - 1}+d[/tex]
In the first choice: a₁ = -23 and
[tex]a_n=a_{n-1}+12[/tex]
Then by recursive formula, it suggests arithmatic sequence with first term -23 and common difference 12.
and
[tex]a_n=-23+12(n-1)[/tex]
So by explicit formula it also suggests arithmatic sequence with first term -23 and common difference 12.
So it is one correct choice.
In second choice: a₁ = -100 and
[tex]a_n=a_{n-1}+1[/tex]
By recursive formula, it suggests arithmatic sequence with first term -100 and common difference 1.
[tex]a_n=100-1(n-1)[/tex]
By explicit formula, it suggests arithmatic sequence with first term 100 and common difference -1.
So it is not the correct choice.
In third choice: a₁ = -41 and
[tex]a_n=a_{n-1}+12[/tex]
By recursive formula, it suggests arithmatic sequence with first term -41 and common difference 12.
[tex]a_n=12-41(n-1)[/tex]
By explicit formula, it suggests arithmatic sequence with first term 12 and common difference -41.
So it is not the correct choice.
In fourth choice: a₁ = 50 and
[tex]a_n=a_{n-1}-4[/tex]
By recursive formula, it suggests arithmatic sequence with first term 50 and common difference -4.
[tex]a_n=50+4(n-1)[/tex]
By explicit formula, it suggests arithmatic sequence with first term 50 and common difference 4.
So it is not the correct choice.
Hence the option (A), the first choice is correct.
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