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1.

The explicit rule for a sequence is given. 

an=1/2(4/3)^n−1

What is the recursive rule for the geometric sequence.

a1= _____   an=  ______

 

 

& 2.

Enter the explicit rule for the geometric sequence.

15,3,3/5,3/25,…

an= 

 

Once again, thanks for your help!

 Apr 5, 2018
 #1
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1. a1 \(-{1\over3}\) by substituting 1 into the explicit rule.

   To find the recursive rule for the sequence, note that  \(a_{n-1}={1\over2}({4\over3})^{n-1}-1\).  Therefore:

 

\({4\over3}a_{n-1}={1\over2}({4\over3})^n-{4\over3}\)

\({4\over3}a_{n-1}+{1\over3}={1\over2}({4\over3})^n-1\)

\({4\over3}a_{n-1}+{1\over3}=a_n\)

 

   To check: \(a_1=-{1\over3}\)\(a_2={1\over2}({4\over3})^2-1={4\over3}a_1+{1\over3}=-{1\over9}\)

 

2. Since the sequence starts at 15 and the common ratio is \(1\over3\), the explicit rule is \(15\over5^{n-1}\).

 Apr 5, 2018

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