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I have two arithmetic sequences. The first term of the first sequence is $0$. The second term of the first sequence is the first term of the first sequence plus the first term of the second sequence. Similarly, the third term of the first sequence is the second term of the first sequence plus the second term of the second sequence. If the fifth term of the second sequence is $3$, what is the fifth term of the first sequence?

Guest Sep 28, 2017
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I've looked at this one for a bit......I'm not sure that the second sequence is an arithmetic one......more like a "constant" one

 

Here's my reasoning

 

If the second term of the first sequence is the first term of the first sequence plus the first term of the second sequence.....since the first term of the first sequence is 0, the second term must just be the common difference between terms of that series.....call it "d"

 

But this means that the first term of the second sequence must be "d"

 

Let the common difference of the second sequence be "e"

 

So.....since the 5th term of this sequence is 3, then the first term, d,  must equal  3  - 4e....so  d  =  3 - 4e  (1)

 

And the third term of the first sequence is the second term of the first sequence, d, plus the second term of the second sequence, 3 - 3e....so....the third term of the first sequence, 2d, must equal d  + ( 3 - 3e)   =  2d.......so......subbing (1)  into this we have that   (3 - 4e) + (3 - 3e)   =  2d  ⇒    6 - 7e  = 2 (3 - 4e)  ⇒    6  - 7e  = 6 - 8e    which implies that  e = 0

 

 

So.........the second sequence must be constant....and the 5th term of the first series must be 0 + 4(d)  =

0 + 4(3)  =  12

 

Here are the two  sequences

 

0    3    6      9      12

3    3    3      3       3

 

Anyone else have any other thoughts  ??

 

cool cool cool

CPhill  Sep 29, 2017

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