Question

Identify the sequence as arithmetic or geometric, and write a recursive formula for the sequence. Be

sure to identify your starting value
14, 21, 28, 35, …

Answer 1

Answer: The sequence is an arithmetic sequence

The recursive formula is given as
`a_{n` =
`a_(n-1)` + 7 , where
`a_(1)` = 14

Explanation:

`\\`For a sequence to be geometric then there must exist a common ratio. Let r represent the common ratio. The formula for calculating common ratio implies:

`\\`r =
`(T_(2) )/(T_(1) )` =
`(T_(3) )/(T_(2) )` … , that is r is calculated by dividing the second term by the first term or the third term divided by the second term and so on.

`\\`To check if the sequence is geometric, let us find the common ratio.

`\\`
`T_(1)` = 14

`\\`
`T_(2)` = 21

`\\`
`T_(3)` = 28

`\\`
`T_(4)` = 35

`\\`So,
`(T_(2) )/(T_(1) )` =
`(21)/(14)` =
`(3)/(2)`

`\\`
`(T_(3) )/(T_(2) )`=
`(28)/(21)` =
`(4)/(3)`

`\\`
`(T_(4) )/(T_(3) )` =
`(35)/(28)` =
`(5)/(4)`

`\\`Considering the result, it is clear that it is not a geometric sequence since the ratios are not the same

`\\`Arithmetic Sequence , for a sequence to be arithmetic then there must be an existence of a common difference.That is

`\\`
`T_(2)` –
`T_(1)`=
`T_(3)` –
`T_(2)` =
`T_(4)`–
`T_{}`

`\\`Let us check if the sequence given follow this rule

`\\`
`T_(2)` –
`T_(1)` = 21 -14 = 7

`\\`
`T_(3)` –
`T_(2)` = 28 – 21 = 7

`\\`
`T_(4)` –
`T_{}` = 35 – 28 = 7

`\\`Therefore the sequence is an arithmetic sequence.

`\\`To find the recursive formula for the sequence

`\\`
`a_(1)` = 14

`\\`
`a_(n)` =
`a_(n-1)` +d

`\\`
`a_(n)` =
`a_(n-1)` +7