Question

2a.4 assignment
geometric sequences: zeroth term and common ratio
instructions: find the missing values in the sequence then state the common ratio and zeroth term.
sequences
10, 50, _, 1250, _
-8, -16, -32, _, _, -256
28, 112, _, 1792, _
60, 30, _, _, _,
-3, -1, -\frac{1}{3}, -\frac{1}{9}, _, _
common ratio
zeroth term

Explanation:

Step1: Recall geometric - sequence formula

The general form of a geometric sequence is \(a_n=a_0r^n\), where \(a_n\) is the \(n\)th term, \(a_0\) is the zeroth - term, \(r\) is the common ratio, and \(n\) is the term number. To find the common ratio \(r\), we use the formula \(r=\frac{a_{n + 1}}{a_n}\).

Sequence 1: \(-3,-1,-\frac{1}{3},-\frac{1}{9}\)

Step1: Calculate common ratio

\(r=\frac{-1}{-3}=\frac{1}{3}\). The zeroth - term \(a_0=-3\div\frac{1}{3}=-9\). The next terms are \(-\frac{1}{9}\times\frac{1}{3}=-\frac{1}{27}\), \(-\frac{1}{27}\times\frac{1}{3}=-\frac{1}{81}\).

Sequence 2: \(60,30\)

Step1: Calculate common ratio

\(r = \frac{30}{60}=\frac{1}{2}\). The zeroth - term \(a_0=60\div\frac{1}{2}=120\). The next terms are \(30\times\frac{1}{2}=15\), \(15\times\frac{1}{2}=7.5\).

Sequence 3: \(28,112\)

Step1: Calculate common ratio

\(r=\frac{112}{28}=4\). The zeroth - term \(a_0=28\div4 = 7\). The next terms are \(112\times4 = 448\), \(448\times4=1792\) (already given), \(1792\times4 = 7168\).

Sequence 4: \(-8,-16,-32\)

Step1: Calculate common ratio

\(r=\frac{-16}{-8}=2\). The zeroth - term \(a_0=-8\div2=-4\). The next terms are \(-32\times2=-64\), \(-64\times2=-128\), \(-128\times2=-256\) (already given).

Sequence 5: \(10,50\)

Step1: Calculate common ratio

\(r=\frac{50}{10}=5\). The zeroth - term \(a_0=10\div5 = 2\). The next terms are \(50\times5=250\), \(250\times5 = 1250\) (already given), \(1250\times5=6250\).

Answer:

Sequence	Zeroth Term	Common Ratio	Next Two Terms
\(60,30\)	\(120\)	\(\frac{1}{2}\)	\(15,7.5\)
\(28,112\)	\(7\)	\(4\)	\(448,7168\)
\(-8,-16,-32\)	\(-4\)	\(2\)	\(-64,-128\)
\(10,50\)	\(2\)	\(5\)	\(250,6250\)