Answer:
[tex]-1,4,-7,10,...[/tex] neither
[tex]192,24,3,\frac{3}{8},...[/tex] geometric progression
[tex]-25,-18,-11,-4,...[/tex] arithmetic progression
Step-by-step explanation:
Given:
sequences: [tex]-1,4,-7,10,...[/tex]
[tex]192,24,3,\frac{3}{8},...[/tex]
[tex]-25,-18,-11,-4,...[/tex]
To find: which of the given sequence forms arithmetic progression, geometric progression or neither of them
Solution:
A sequence forms an arithmetic progression if difference between terms remain same.
A sequence forms a geometric progression if ratio of the consecutive terms is same.
For [tex]-1,4,-7,10,...[/tex]:
[tex]4-(-1)=5\\-7-4=-11\\10-(-7)=17\\So,\,\,4-(-1)\neq -7-4\neq 10-(-7)[/tex]
Hence,the given sequence does not form an arithmetic progression.
[tex]\frac{4}{-1}=-4\\\frac{-7}{4}=\frac{-7}{4}\\\frac{10}{-7}=\frac{-10}{7}\\So,\,\,\frac{4}{-1}\neq \frac{-7}{4}\neq \frac{10}{-7}[/tex]
Hence,the given sequence does not form a geometric progression.
So, [tex]-1,4,-7,10,...[/tex] is neither an arithmetic progression nor a geometric progression.
For [tex]192,24,3,\frac{3}{8},...[/tex] :
[tex]\frac{24}{192}=\frac{1}{8}\\\frac{3}{24}=\frac{1}{8}\\\frac{\frac{3}{8}}{3}=\frac{1}{8}\\So,\,\,\frac{24}{192}=\frac{3}{24}=\frac{\frac{3}{8}}{3}[/tex]
As ratio of the consecutive terms is same, the sequence forms a geometric progression.
For [tex]-25,-18,-11,-4,...[/tex] :
[tex]-18-(-25)=-18+25=7\\-11-(-18)=-11+18=7\\-4-(-11)=-4+11=7\\So,\,\,-18-(-25)=-11-(-18)=-4-(-11)[/tex]
As the difference between the consecutive terms is the same, the sequence forms an arithmetic progression.
