lavalava
The fourth term of an arithmetic sequence is 141, and the seventh term is 132. The first term is _____.
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anonymous
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\[a, a+d,a+2d,a+3d,a+4d,...\] you got \(a+3d=141\) and \(a+6d=132\)
anonymous
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that means \(132-141=(a+6d)-(a+3d)=3d=-9\) and so \(d=-3\)
lavalava
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okay so then d=4?
anonymous
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actually \(d=-3\)
lavalava
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umm please explain!!
anonymous
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k lets go slow
lavalava
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please and thanks
anonymous
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actually before we even start, since \(132<141\) is it clear that the terms are getting smaller?
anonymous
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in other words, "\(d\)" the "common difference" must be negative right?
lavalava
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okay
anonymous
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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
anonymous
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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
lavalava
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okay
anonymous
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The fourth term of an arithmetic sequence is 141
tells you that \(a+3d=141\)
lavalava
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okay
anonymous
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you see that it is the fourth term, so it is \(a+3d\) not \(a+4d\)
lavalava
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ohhh
anonymous
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and the seventh term is 132 means
\[a+6d=132\]
anonymous
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from these two pieced of information we can solve for \(d\) and then solve for \(a\)
anonymous
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*pieces
lavalava
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okay
anonymous
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a bit of algebra shows that
\[a+6d-(a+3d)=3d\] right?
lavalava
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ohh okay i get it..
anonymous
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so we see that
\[3d=132-141=-9\]
lavalava
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mhm
anonymous
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so far so good?
lavalava
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and so since it is 7-4= 3 then -9/3 would be -3 right? giving us the difference
anonymous
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yeah \(-3\) is the difference
anonymous
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what you said
lavalava
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ohhh okay!!! i get it!! :D
anonymous
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you are still not done though right?
anonymous
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your question asked "The first term is _____"
lavalava
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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
anonymous
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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 \(a_1\)
lavalava
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:D yay thank you!!! :D
ohh okay... ill keep in mind that equation!! thank you soo much!
anonymous
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yw