How do you write the first five terms of the geometric sequence $a_{1} = 6, a_{k + 1} = - \frac{3}{2} a_{k}$ $a_1=6, a_(k+1)=-3/2a_k$ and determine the common ratio and write the nth term of the sequence as a function of n?

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How do you write the first five terms of the geometric sequence $a_{1} = 6, a_{k + 1} = - \frac{3}{2} a_{k}$ $a_1=6, a_(k+1)=-3/2a_k$ and determine the common ratio and write the nth term of the sequence as a function of n?

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Answer 1 · 2018-03-08T13:05:07

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Explanation:

A geometric sequence is defined as: an ordered sequence of number such that each term is calculated multipliying the prior term by a constant number called ratio. In mathematical terms

$a_{k + 1} = r \cdot a_{k}$ $a_(k+1)=r·a_k$ or if you want $a_{k} = r \cdot a_{k - 1}$ $a_k=r·a_(k-1)$

In our case $a_{k + 1} = - \frac{3}{2} \cdot a_{k}$ $a_(k+1)=-3/2·a_k$ . So, the ratio is $- \frac{3}{2}$ $-3/2$ .

Lets calculate:

$a_{1} = 6$ $a_1=6$
$a_{2} = - \frac{3}{2} \cdot 6 = - 9$ $a_2=-3/2·6=-9$
$a_{3} = - \frac{3}{2} \cdot (- 9) = + \frac{27}{2}$ $a_3=-3/2·(-9)=+27/2$
$a_{4} = - \frac{3}{2} \cdot \frac{27}{2} = - \frac{81}{4}$ $a_4=-3/2·27/2=-81/4$
$a_{5} = - \frac{3}{2} \cdot (- \frac{81}{4}) = + \frac{243}{8}$ $a_5=-3/2·(-81/4)=+243/8$

The general term is given by $a_{n} = a_{1} \cdot r^{n - 1}$ $a_n=a_1·r^(n-1)$ ...in our case

$a_{n} = 6 \cdot {(- \frac{3}{2})}^{n - 1}$ $a_n=6·(-3/2)^(n-1)$

Each term with a odd position will be positive and each term in a even postion will be negative (oscillating)

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How do you write the first five terms of the geometric sequence $a_{1} = 6, a_{k + 1} = - \frac{3}{2} a_{k}$ $a_1=6, a_(k+1)=-3/2a_k$ and determine the common ratio and write the nth term of the sequence as a function of n?

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How do you write the first five terms of the geometric sequence a_1=6, a_(k+1)=-3/2a_ka1=6,ak+1=−32aka_1=6, a_(k+1)=-3/2a_k and determine the common ratio and write the nth term of the sequence as a function of n?

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Explanation:

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How do you write the first five terms of the geometric sequence $a_{1} = 6, a_{k + 1} = - \frac{3}{2} a_{k}$ $a_1=6, a_(k+1)=-3/2a_k$ and determine the common ratio and write the nth term of the sequence as a function of n?