Question

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mr. jones asks his students to generate the next two numbers in the sequence beginning −5.5, 11, … . taquan suggests that the sequence is geometric and the next two numbers are −22 and 44. julia suggests that the sequence is arithmetic and the next two numbers are 27.5 and 44. which best explains which student is correct?

first box
both students could be correct about the types of possible sequences. however, one student made a computational error because it is not possible to arrive at a fourth term of 44 in two different ways.

second box
julia is correct. when the numbers alternate between decimals and whole numbers, the sequence is arithmetic. each successive term is generated by adding 16.5.

third box
both students could be correct. because two numbers are given in the original sequence, it is possible to find a common difference and common ratio between the successive terms.

Explanation:

Step1: Analyze Taquan's claim (geometric sequence)

For a geometric sequence, the common ratio \( r \) is \( \frac{a_{n + 1}}{a_n} \). Given \( a_1=-5.5 \), \( a_2 = 11 \), the ratio \( r=\frac{11}{-5.5}=-2 \). Then \( a_3=a_2\times r=11\times(-2)= - 22 \), \( a_4=a_3\times r=-22\times(-2)=44 \). So Taquan's next two terms (-22, 44) are correct for geometric.

Step2: Analyze Julia's claim (arithmetic sequence)

For an arithmetic sequence, the common difference \( d=a_{n + 1}-a_n \). \( d = 11-(-5.5)=16.5 \). Then \( a_3=a_2 + d=11 + 16.5 = 27.5 \), \( a_4=a_3 + d=27.5+16.5 = 44 \). So Julia's next two terms (27.5, 44) are correct for arithmetic.

Step3: Evaluate the explanations

• First explanation: Says one student made an error, but both are correct. So this is wrong.
• Second explanation: Julia's sequence alternates decimals and whole? No, -5.5 (decimal), 11 (whole), 27.5 (decimal), 44 (whole) – but the key is the sequence can be arithmetic (common difference 16.5) and geometric (common ratio -2) both leading to 44 as 4th term. Wait, no, the third explanation: "Both students could be correct. Because two numbers are given in the original sequence, it is possible to find a common difference and common ratio between the successive terms." Let's check: For the two given terms (-5.5, 11), common difference \( d = 16.5 \), common ratio \( r=-2 \). So with two terms, we can define an arithmetic (using \( d \)) or geometric (using \( r \)) sequence, both valid, and both can reach 44 as the 4th term (as shown in step1 and step2). The second explanation's reasoning about alternating decimals is not the main reason. The third explanation correctly states that with two terms, we can have both arithmetic (common difference) and geometric (common ratio) sequences, so both students can be correct.

Answer:

The explanation "Both students could be correct. Because two numbers are given in the original sequence, it is possible to find a common difference and common ratio between the successive terms." (the third boxed explanation) is correct.