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Which of the following recursive formulas represents the same arithmetic sequence as the explicit formula an = 2 + (n โ€“ 1)5?A.{๐‘Ž1=5๐‘Ž๐‘›=(๐‘Ž๐‘›โˆ’1+5)2{ a 1โ€‹ =5a nโ€‹ =(a nโˆ’1โ€‹ +5)2โ€‹ B.{๐‘Ž1=2๐‘Ž๐‘›=๐‘Ž๐‘›โˆ’1โˆ™5{ a 1โ€‹ =2a nโ€‹ =a nโˆ’1โ€‹ โˆ™5โ€‹ C.{๐‘Ž1=5๐‘Ž๐‘›=๐‘Ž๐‘›โˆ’1+2{ a 1โ€‹ =5a nโ€‹ =a nโˆ’1โ€‹ +2โ€‹ D.{๐‘Ž1=2๐‘Ž๐‘›=๐‘Ž๐‘›โˆ’1+5{ a 1โ€‹ =2a nโ€‹ =a nโˆ’1โ€‹ +5โ€‹

Question

Which of the following recursive formulas represents the same arithmetic sequence as the explicit formula an = 2 + (n โ€“ 1)5?A.{๐‘Ž1=5๐‘Ž๐‘›=(๐‘Ž๐‘›โˆ’1+5)2{ a 1โ€‹ =5a nโ€‹ =(a nโˆ’1โ€‹ +5)2โ€‹ B.{๐‘Ž1=2๐‘Ž๐‘›=๐‘Ž๐‘›โˆ’1โˆ™5{ a 1โ€‹ =2a nโ€‹ =a nโˆ’1โ€‹ โˆ™5โ€‹ C.{๐‘Ž1=5๐‘Ž๐‘›=๐‘Ž๐‘›โˆ’1+2{ a 1โ€‹ =5a nโ€‹ =a nโˆ’1โ€‹ +2โ€‹ D.{๐‘Ž1=2๐‘Ž๐‘›=๐‘Ž๐‘›โˆ’1+5{ a 1โ€‹ =2a nโ€‹ =a nโˆ’1โ€‹ +5โ€‹

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Solution

The explicit formula given is an = 2 + (n โ€“ 1)5. This represents an arithmetic sequence where the first term (a1) is 2 and the common difference (d) is 5.

Looking at the options:

A. {๐‘Ž1=5๐‘Ž๐‘›=(๐‘Ž๐‘›โˆ’1+5)2: This is incorrect because the first term is not 2 and the common difference is not 5.

B. {๐‘Ž1=2๐‘Ž๐‘›=๐‘Ž๐‘›โˆ’1โˆ™5: This is incorrect because the common difference is not multiplication, it should be addition.

C. {๐‘Ž1=5๐‘Ž๐‘›=๐‘Ž๐‘›โˆ’1+2: This is incorrect because the first term is not 2 and the common difference is not 5.

D. {๐‘Ž1=2๐‘Ž๐‘›=๐‘Ž๐‘›โˆ’1+5: This is correct because the first term is 2 and the common difference is 5.

So, the answer is D. {๐‘Ž1=2๐‘Ž๐‘›=๐‘Ž๐‘›โˆ’1+5.

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