# 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.

Jun 6, 2018

${t}_{n} = - 4 n + 91$

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

#### Explanation:

Each term is $4$ less than the previous term so the sequence is linear and the ${n}^{t h}$ term will have something to do with $- 4 n$

sequence$\text{ "87," "83," " 79," } 75 \ldots .$
$- 4 n \implies \text{ " -4," " -8," " -12," } - 16$

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

So the ${n}^{t h}$ term is ${T}_{n} = - 4 n + 91$

To see if a number is in the sequence, equate the ${n}^{t h}$ term to the number and solve it for $n$. If we get a whole number answer then it is in the sequence.

$- 4 n + 91 = - 145$

subtract $91$ from both sides

$- 4 n = - 145 - 91$

$- 4 n = 236$

divide by $- 4$

$n = 59$

Term number $59$ in the sequence

Jun 6, 2018

${t}_{n} = - 4 n + 91$

${t}_{59} = - 145$

#### Explanation:

In sequence $87 , 83 , 79 , 75 , \ldots$ 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:

${t}_{n} = {t}_{1} + \left(n - 1\right) d$...in our case

${t}_{n} = 87 + \left(n - 1\right) \left(- 4\right)$

${t}_{n} = 87 - 4 n + 4 = - 4 n + 91$

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

$4 n = 91 + 145 = 236$

$n = \frac{236}{4} = 59$

Thus, the ${59}^{t h}$ term is $- 145$