Question

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? taquan is correct. when the signs change in a sequence, the sequence is geometric. each successive term is generated by multiplying by -2. 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. 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. 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: Check Taquan's claim (geometric sequence)

For a geometric sequence, the common ratio \(r=\frac{a_{n + 1}}{a_{n}}\). Given \(a_1=-5\) and \(a_2 = 11\), \(r=\frac{11}{-5}=- 2.2\). If it were geometric with \(r=-2\) (Taquan's multiplier), \(a_2=-5\times(-2) = 10
eq11\). But if we consider the general concept of a geometric sequence (not Taquan's wrong - ratio calculation), a geometric sequence is defined by \(a_{n+1}=a_{n}\times r\). Also, if we assume a geometric sequence with \(a_1=-5,a_2 = 11,a_3=-22,a_4 = 44\), \(r=\frac{11}{-5}=\frac{-22}{11}=\frac{44}{-22}=-2.2\) (but if we just go by the multiplier of \(-2\) for the sake of term - generation from the first two terms \(a_1=-5,a_2 = 11\) is wrong, but the idea of a geometric sequence is possible).

Step2: Check Julia's claim (arithmetic sequence)

For an arithmetic sequence, the common difference \(d=a_{n + 1}-a_{n}\). If \(a_1=-5,a_2 = 11\), \(d=11-(-5)=16\). If \(d = 16.5\) (Julia's common difference), \(a_2=-5 + 16.5=11.5
eq11\). But the idea of an arithmetic sequence is defined by \(a_{n+1}=a_{n}+d\). If \(a_1=-5,a_2 = 11,a_3=27.5,a_4 = 44\), \(d=11-(-5)=16\) (wrong for Julia's \(d = 16.5\) when calculating from \(a_1\) to \(a_2\)), but the concept of an arithmetic sequence is possible.

Step3: Consider the nature of sequences

A sequence is geometric if \(a_{n + 1}=a_{n}\times r\) (where \(r\) is a non - zero constant) and arithmetic if \(a_{n+1}=a_{n}+d\) (where \(d\) is a constant). Given only two terms \(a_1\) and \(a_2\) of a sequence, we can assume different rules (common ratio \(r\) or common difference \(d\)) to extend the sequence.

Answer:

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.