In an arithmetic sequence, the first term is -2, the fourth term is 16, and the nth term is 11998, how do you find n and the common difference?

Question

5391 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-05-12T14:16:17

common difference, $d=6$

$n=2001$

$a_n = a + (n-1)d$

Here, $a$ is the first term, $d$ is the common difference and $a_n$ is the nth term of the sequence.

We are given:

$color(red)("To find "n" and " d.)$

Let's start with finding $d$ :

$a_n = a + (n-1)d$

$color(brown )("Put in "a=-2 " and " n=4)$

$a_4 = -2 + (4-1)d$

$16 = -2 + (4-1)d$ , $color(blue)(" since " a_4=16)$

$16 = -2 + 3d$

Add $2$ to both sides:

$16 color(blue)(+ 2) = -2 color(blue)(+ 2)+ 3d$

$18 = 3d$

$color(red)(6 = d )$

Next, calculate $n$ :

$a_n = a + (n-1)d$

$color(brown)("Put in "a_n=11998 " , " a=-2 " and " d=6)$

$11998 = -2 + (n-1)6$

Add $2$ to both sides:

$11998 color(blue)(+ 2) = -2 color(blue)(+ 2) + (n-1)6$

$12000 = (n-1)6$

Divide both sides by $6$ :

$12000/ color(blue)6 = [(n-1)6]/ color(blue)6$

$2000 = n-1$

Add $1$ to both sides:

$2000 color(blue)(+ 1) = n-1 color(blue)(+ 1)$

$color(red)(2001 = n)$