Question

The fourth term of an arithmetic sequence is 141, and the seventh term is 132. The first term is _____.

Answer 1

\[a, a+d,a+2d,a+3d,a+4d,...\] you got \(a+3d=141\) and \(a+6d=132\)

Answer 2

that means \(132-141=(a+6d)-(a+3d)=3d=-9\) and so \(d=-3\)

Answer 3

okay so then d=4?

Answer 4

actually \(d=-3\)

Answer 5

umm please explain!!

Answer 6

k lets go slow

Answer 7

please and thanks

Answer 8

actually before we even start, since \(132<141\) is it clear that the terms are getting smaller?

Answer 9

in other words, "\(d\)" the "common difference" must be negative right?

Answer 10

okay

Answer 11

if we call the first term \(a\) then the second term is \(a+d\) for some \(d\) and the third term is \(a+2d\), the fourth term is \(a+3d\) the fifth term is \(a+4d\)
etc

Answer 12

in other words, you keep adding \(d\) to each term to get the next term, with the sophistication that you might be "adding" a negative number

Answer 13

okay

Answer 14

The fourth term of an arithmetic sequence is 141
tells you that \(a+3d=141\)

Answer 15

okay

Answer 16

you see that it is the fourth term, so it is \(a+3d\) not \(a+4d\)

Answer 17

ohhh

Answer 18

and the seventh term is 132 means
\[a+6d=132\]

Answer 19

from these two pieced of information we can solve for \(d\) and then solve for \(a\)

Answer 20

*pieces

Answer 21

okay

Answer 22

a bit of algebra shows that
\[a+6d-(a+3d)=3d\] right?

Answer 23

ohh okay i get it..

Answer 24

so we see that
\[3d=132-141=-9\]

Answer 25

mhm

Answer 26

so far so good?

Answer 27

and so since it is 7-4= 3 then -9/3 would be -3 right? giving us the difference

Answer 28

yeah \(-3\) is the difference

Answer 29

what you said

Answer 30

ohhh okay!!! i get it!! :D

Answer 31

you are still not done though right?

Answer 32

your question asked "The first term is _____"

Answer 33

hmm well pluggin in the difference and then your equation...
a4=141
a3=141+3=144
a2=144+3=147
a1=147+3=150
so then the first term would be 150

right? i just did it backwards

Answer 34

yeah i guess so
i would have said \(a+3\times (-3)=141\)or
\[a-9=141\] making \(a=150\)
your method means you understand what is going on, which is good
but you certainly wouldn't want to use that if you had say \(a_{75}\) and wanted \(a_1\)

Answer 35

:D yay thank you!!! :D
ohh okay... ill keep in mind that equation!! thank you soo much!

Answer 36

yw