Question

Given the recursive formula for an arithmetic sequence find the common difference and the 52nd term

Answer 1

Common difference(d)
`a_(52)`

(21) -10 -548

(22) -7 -323

(23) 10 547

(24) -100 -5118

Explanation:

Let the common difference be denoted by 'd'.

Also the nth difference of an arithmetic sequence is given by:
`a_(n)=a_(1)+(n-1)* d`

(21)

We are given a recursive formula as:

`a_(n)=a_(n-1)-10`

The first term is given by:

`a_(1)=-38`

The common difference for an arithmetic sequence is given by:

`a_(n)-a_(n-1)`

Hence, here we have the common difference as:

`d=-10`

The nth term of an arithmetic sequence is given by:

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

Here
`a_(1)=-38` and
`d=-10`.

Hence,
`a_(52)=-38+(52-1)* (-10)`

Hence,
`a_(52)=-548`

(22)

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

`a_(1)=34`

The common difference for an arithmetic sequence is given by:

`a_(n)-a_(n-1)`

Hence, here we have the common difference as:

`d=-7`

Here
`a_(1)=34` and
`d=-7`.

Hence,
`a_(52)=34+(52-1)* (-7)`

Hence,
`a_(52)=-323`

(23)

`a_(n)=a_(n-1)+10`

`a_(1)=37`

The common difference for an arithmetic sequence is given by:

`a_(n)-a_(n-1)`

Hence, here we have the common difference as:

`d=10`

Here
`a_(1)=37` and
`d=10`.

Hence,
`a_(52)=37+(52-1)* (10)`

Hence,
`a_(52)=547`

(24)

`a_(n)=a_(n-1)-100`

`a_(1)=-18`

The common difference for an arithmetic sequence is given by:

`a_(n)-a_(n-1)`

Hence, here we have the common difference as:

`d=-100`

Here
`a_(1)=-18` and
`d=-100`.

Hence,
`a_(52)=-18+(52-1)* (-100)`

Hence,
`a_(52)=-5118`