Question

1. Algebra: Consider the arithmetic sequence 87, 83, 79, 75, . . .. (You may assume that the pattern displayed continues for the whole sequence.)?

(a) Find the formula for the general term tn.
(b) Is −145 a member of the sequence. Give reasons for your answer.

Answer 1

`t_n=-4n+91`

`-145` is term `59` in the sequence

Explanation:

Each term is `4` less than the previous term so the sequence is linear and the `n^(th)` term will have something to do with `-4n`

sequence`" "87," "83," " 79," " 75 ....`
`-4n =>" " -4," " -8," " -12," " -16`

The difference between the sequence and the `-4n` is `91` each time

So the `n^(th)` term is `T_n =-4n+91`

To see if a number is in the sequence, equate the `n^(th)` term to the number and solve it for `n`. If we get a whole number answer then it is in the sequence.

`-4n+91=-145`

subtract `91` from both sides

`-4n=-145-91`

`-4n = 236`

divide by `-4`

`n=59`

Term number `59` in the sequence

Answer 2

`t_n =-4n+91`

`t_59 = -145`

Explanation:

In sequence `87,83,79,75,...` we observe that

`83=87-4`
`79=83-4` and so on

Then, is an arithmetic sequence of difference `-4` and first term `87`

a) We know that general term in an arithmetic sequence is given by:

`t_n=t_1+(n-1)d`...in our case

`t_n=87+(n-1)(-4)`

`t_n =87-4n+4=-4n+91`

b) let's see if there is an `n` such that `-145=-4n+91`

`4n=91+145=236`

`n=236/4=59`

Thus, the `59^(th)` term is `-145`