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!
+349 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 \(15\over5^{n-1}\).
