Question

Question 8 is find a closed form for the infinite arithmetic sequence ?

Answer 1

`x_n=3/2n+1/2`

Explanation:

`"the nth term of an arithmetic sequence is"`

`•color(white)(x)a_n=a+(n-1)d`

`"We require to find d the "color(blue)"common difference"`

`•color(white)(x)d=a_2-a_1=a_3-a_2= ...... =a_n-a_(n-1)`

`"Generate the terms of the sequence using the recurrence"`
`"relation"`

`a_1=a=2larrcolor(red)"given"`

`a_2=2+3/2=7/2`

`a_3=7/2+3/2=5`

`a_4=5+3/2=13/2`

`a_5=13/2+3/2=8`

`"the arithmetic sequence is "`

`2,7/2,5,13/2,8,....`

`"with "a=2" and "d=7/2-2=5-7/2=3/2`

`rArrx_n=2+3/2(n-1)larrcolor(blue)"n th term formula"`

`color(white)(rArrx_n)=2+3/2n-3/2`

`color(white)(rArrx_n)=3/2n+1/2`

`color(blue)"As a check"`

`n=5tox_5=(3/2xx5)+1/2=15/2+1/2=8" True"`

`color(blue)"NOTE"`

`"A recurrence relation allows us to find a term in the "`
`"sequence if we know the previous term whereas the"`
`"nth term formula (closed form) allows us to find any"`
`"term in the sequence given it's position n in the"`
`"sequence"`