Question

assignment 9.1 sequences and their nota
due wednesday by 11:59pm points 8 submitting an external to
assignment 9.1 sequences and their notation
score: 4/8 answered: 4/8
question 5
find a formula for the general term ( a_{n} ) of the sequence assuming the pattern of
continues.
\\( \left\\{ - \frac { 5 } { 2 } , \frac { 7 } { 4 } , - \frac { 9 } { 8 } , \frac { 11 } { 16 } , - \frac { 13 } { 32 } , \ldots \
ight\\} \\)
assume the first term is ( a _ { 1 } )
\\( a _ { n } = \\)
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Explanation:

Step1: Analyze the sign pattern

The sign of the terms alternates. For \(n = 1\), the sign is negative, for \(n=2\) positive, etc. We can represent the sign using \((- 1)^{n}\).

Step2: Analyze the numerator pattern

The numerators are \(5,7,9,11,13,\cdots\). This is an arithmetic sequence with first term \(a_{1}=5\) and common difference \(d = 2\). The formula for the \(n\)th term of an arithmetic sequence \(a_{n}=a_{1}+(n - 1)d\). Substituting \(a_{1}=5\) and \(d = 2\), we get \(a_{n}=5+(n - 1)\times2=2n + 3\).

Step3: Analyze the denominator pattern

The denominators are \(2,4,8,16,32,\cdots\). This is a geometric sequence with first term \(b_{1}=2\) and common ratio \(r = 2\). The formula for the \(n\)th term of a geometric sequence \(b_{n}=b_{1}\times r^{n-1}\). Substituting \(b_{1}=2\) and \(r = 2\), we get \(b_{n}=2\times2^{n - 1}=2^{n}\).

Answer:

\(a_{n}=(-1)^{n}\frac{2n + 3}{2^{n}}\)