# How do you find an equation that describes the sequence 7, 10, 13, 16,... and find the 89th term?

Oct 27, 2017

Arithmatic sequene with common diference $3$ and
$89$ th term is $271$

#### Explanation:

This is an arithmatic sequene , the common difference is

d=16-13=13-10=10-7=3 ; 1st term is $a = 7$

$n$ th term is ${t}_{n} = a + \left(n - 1\right) d \therefore 89$ th term is

${t}_{89} = 7 + \left(89 - 1\right) 3 = 7 + 264 = 271$

$89$ th term is $271$ [Ans]

Oct 27, 2017

${a}_{n} = 3 n + 4 , {a}_{89} = 271$

#### Explanation:

$\text{the given terms are the terms of an "color(blue)"arithmetic sequence}$

$a , a + d , a + 2 d , a + 3 d , \ldots \ldots , a + \left(n - 1\right) d$

$\text{where a is the first term and d the "color(blue)"common difference}$

$d = {a}_{2} - {a}_{1} = {a}_{3} - {a}_{2} = \ldots \ldots = {a}_{n} - {a}_{n - 1}$

$\text{here } d = 10 - 7 = 13 - 10 = 16 - 13 = 3$

$\text{the nth term of the sequence is}$

•color(white)(x)a_n=a+(n-1)d

$\Rightarrow {a}_{n} = 7 + 3 \left(n - 1\right) = 7 + 3 n - 3$

$\Rightarrow {a}_{n} = 3 n + 4 \leftarrow \textcolor{red}{\text{nth term formula for sequence}}$

$\Rightarrow {a}_{89} = \left(3 \times 89\right) + 4 = 271$