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?

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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) #