Question

Determine whether each of the following sequences are arithmetic, geometric or neither. If arithmetic, state the common difference. If geometric, state the common ratio.

4, 13/3, 14/3, 5, 16/3, …

Is the answer: Neither?

Answer 1

Arithmetic with common difference of
`\sf (1)/(3)`

Explanation:

`\textsf{Given sequence}=4, (13)/(3), (14)/(3), 5, (16)/(3),...`

If a sequence is arithmetic, the difference between consecutive terms is the same (this is called the common difference).

If a sequence is geometric, the ratio between consecutive terms is the same (this is called the common ratio).

`\sf 4\quad \overset{+(1)/(3)}{\longrightarrow}\quad(13)/(3)\quad \overset{+(1)/(3)}{\longrightarrow}\quad (14)/(3)\quad \overset{+(1)/(3)}{\longrightarrow}\quad 5\quad \overset{+(1)/(3)}{\longrightarrow}\quad (16)/(3)`

As the difference between consecutive terms is
`\sf (1)/(3)` then the sequence is arithmetic with common difference of
`\sf (1)/(3)`

General form of an arithmetic sequence:
`\sf a_n=a+(n-1)d`

where:

• 
  `\sf a_n` is the nth term
• a is the first term
• d is the common difference between terms

Given:

• a = 4
• 
  `\sf d=(1)/(3)`

So the formula for the nth term of this sequence is:

`\implies \sf a_n=4+(n-1)(1)/(3)`

`\implies \sf a_n=(1)/(3)n+(11)/(3)`

Answer 2

`\qquad\qquad\huge\underline{{\sf Answer}}`

`\textbf{Let's see if the sequence is Arithmetic or Geometric :}`

`\textsf{If the difference between successive terms is }`
`\textsf{equal then, the terms are in AP}`

• 
  `\sf{ (14)/(3)- (13)/(3) = (1)/(3)}`

• 
  `\sf{ {5}{}- (14)/(3) = (15-14)/(3) =(1)/(3)}`

`\textsf{Since the common difference is same, }`
`\textsf{we can infer that it's an Arithmetic progression}`
`\textsf{with common difference of } \sf (1)/(3)`