SOLUTION: First Question: Find the initial term of the arithmetic sequence with 10th term -1 and 20th term 7. (use initial term a0) Second Question: There are two geometric sequences w

Algebra ->  Algebra  -> Sequences-and-series -> SOLUTION: First Question: Find the initial term of the arithmetic sequence with 10th term -1 and 20th term 7. (use initial term a0) Second Question: There are two geometric sequences w      Log On

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Question 622864: First Question: Find the initial term of the arithmetic sequence with 10th term -1 and 20th term 7. (use initial term a0)

Second Question: There are two geometric sequences with 2nd term 5 and 4th term 80. What are the two possible values of the 3rd term?

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Answer by MathTherapy(1423) About Me  (Show Source):
You can put this solution on YOUR website!

Using the formula for an arithmetic sequencet%5Bn%5D+=+t%5B1%5D+%2B+%28n+-+1%29d, we get:

t%5B10%5D+=+t%5B1%5D+%2B+%28n+-+1%29d
t%5B10%5D+=+t%5B1%5D+%2B+%2810+-+1%29d ----- Substituting 10 for n in formula
-+1+=+t%5B1%5D+%2B+%2810+-+1%29d
-+1+=+t%5B1%5D+%2B+9d
t%5B1%5D+%2B+9d+=+-+1 ---- eq (i)

t%5Bn%5D+=+t%5B1%5D+%2B+%28n+-+1%29d
t%5B20%5D+=+t%5B1%5D+%2B+%2820+-+1%29d ----- Substituting 20 for n in formula
7+=+t%5B1%5D+%2B+%2820+-+1%29d
7+=+t%5B1%5D+%2B+19d
t%5B1%5D+%2B+19d+=+7 ------ eq (ii)

t%5B1%5D+%2B+9d+=+-+1 ---- eq (i)
t%5B1%5D+%2B+19d+=+7 ---- eq (ii)
- 10d = - 8 ------ Subtracting eq (ii) from eq (i)
d, or difference = %28-+8%29%2F-+10, or 4%2F5

Since d = 4%2F5, then t%5Bn%5D+=+t%5B1%5D+%2B+%28n+-+1%29d becomes: t%5B10%5D+=+t%5B1%5D+%2B+%2810+-+1%29+%2A+%284%2F5%29 ----- Substituting 10 for n, and 4%2F5 for d in formula
-+1+=+t%5B1%5D+%2B+9+%2A+%284%2F5%29
-+1+=+t%5B1%5D+%2B+%2836%2F5%29
t%5B1%5D+=+-+1+-+%2836%2F5+%29
t%5B1%5D+=+-+%285%2F5%29+-+%2836%2F5%29
Initial term, or highlight_green%28t%5B1%5D+=+-+%2841%2F5%29%29

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10th term
t%5Bn%5D+=+t%5B1%5D+%2B+%28n+-+1%29d
-+1+=+%28-+41%2F5%29+%2B+%2810+-+1%29+%2A+%284%2F5%29
-+1+=+%28-+41%2F5%29+%2B+9+%2A+%284%2F5%29
-+1+=+%28-+41%2F5%29+%2B+%2836%2F5%29
-+1+=+-+%285%2F5%29
- 1 = - 1 (TRUE)

20th term
t%5Bn%5D+=+t%5B1%5D+%2B+%28n+-+1%29d
7+=+%28-+41%2F5%29+%2B+%2820+-+1%29+%2A+%284%2F5%29
7+=+%28-+41%2F5%29+%2B+19+%2A+%284%2F5%29
7+=+%28-+41%2F5%29+%2B+%2876%2F5%29
7+=+%2835%2F5%29
7 = 7 (TRUE)

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