the second term of a sequence is 6, and the fourth term is 24. of the following, which statement is true?\na…

the second term of a sequence is 6, and the fourth term is 24. of the following, which statement is true?\na if the sequence is geometric, the first term could be 1.\nb if the sequence is arithmetic, the third term could be 12.\nc if the sequence is geometric, the fifth term could be 48.\nd if the sequence is arithmetic, the sixth term could be 48.

the second term of a sequence is 6, and the fourth term is 24. of the following, which statement is true?\na if the sequence is geometric, the first term could be 1.\nb if the sequence is arithmetic, the third term could be 12.\nc if the sequence is geometric, the fifth term could be 48.\nd if the sequence is arithmetic, the sixth term could be 48.

Answer

Answer:

C. If the sequence is geometric, the fifth term could be 48.

Explanation:

Step1: Analyze arithmetic sequence

For an arithmetic sequence, the formula is (a_{n}=a_{1}+(n - 1)d). Given (a_{2}=a_{1}+d = 6) and (a_{4}=a_{1}+3d=24). Subtract the first equation from the second: ((a_{1}+3d)-(a_{1}+d)=24 - 6), so (2d = 18), (d = 9), (a_{1}=- 3). Then (a_{3}=a_{1}+2d=-3 + 18=15) (not 12), (a_{6}=a_{1}+5d=-3+45 = 42) (not 48).

Step2: Analyze geometric sequence

For a geometric sequence, the formula is (a_{n}=a_{1}r^{n - 1}). Given (a_{2}=a_{1}r = 6) and (a_{4}=a_{1}r^{3}=24). Divide the second equation by the first: (\frac{a_{1}r^{3}}{a_{1}r}=\frac{24}{6}), so (r^{2}=4), (r=\pm2). If (r = 2), then (a_{1}=3); if (r=-2), then (a_{1}=-3). When (r = 2), (a_{5}=a_{4}r=24\times2 = 48). When (r=-2), (a_{5}=a_{4}r=24\times(-2)=-48). So the fifth - term could be 48. If (a_{1}r = 6), when (a_{1}=1), (r = 6), then (a_{4}=a_{1}r^{3}=1\times6^{3}=216\neq24).