Write each of the following series in sigma notation: $$ \begin{array}{ll} ext { (a) } 2+4+6+8+10+12 & ext { (b) } 1+8+27+64+125 \ ext { (c) } \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots .+\frac{1}{128} & ext { (d) } 3-6+12-24+\ldots . ext { to } n ext { terms } \ ext { (e) } \frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\ldots+\frac{24}{25} & ext { (f) } 5+5+5+5+5+\ldots . ext { to } n ext { terms }\end{array} $$

Question

Write each of the following series in sigma notation: $$ \begin{array}{ll}	ext { (a) } 2+4+6+8+10+12 & 	ext { (b) } 1+8+27+64+125 \ 	ext { (c) } \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots .+\frac{1}{128} & 	ext { (d) } 3-6+12-24+\ldots . 	ext { to } n 	ext { terms } \ 	ext { (e) } \frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\ldots+\frac{24}{25} & 	ext { (f) } 5+5+5+5+5+\ldots . 	ext { to } n 	ext { terms }\end{array} $$

UpStudy AI · Accepted Answer

Sure, let's express each of the given series in sigma notation.

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**(a) $$2 + 4 + 6 + 8 + 10 + 12$$**

This is an arithmetic series where each term increases by 2. The $$n$$-th term can be represented as $$2n$$.

$$
\sum_{n=1}^{6} 2n
$$

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**(b) $$1 + 8 + 27 + 64 + 125$$**

This series consists of the cubes of the first five natural numbers. The $$n$$-th term is $$n^3$$.

$$
\sum_{n=1}^{5} n^{3}
$$

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**(c) $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots + \frac{1}{128}$$**

This is a geometric series where each term is multiplied by $$\frac{1}{2}$$. The general term is $$\left(\frac{1}{2}\right)^n$$, starting from $$n=1$$ to $$n=7$$ since $$\frac{1}{128} = \left(\frac{1}{2}\right)^7$$.

$$
\sum_{n=1}^{7} \left(\frac{1}{2}\right)^n
$$

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**(d) $$3 - 6 + 12 - 24 + \ldots$$ to $$n$$ terms**

This is a geometric series with alternating signs. The general term can be written as $$3 \times (-2)^{n-1}$$.

$$
\sum_{k=1}^{n} 3 \cdot (-2)^{k-1}
$$

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**(e) $$\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \ldots + \frac{24}{25}$$**

This is a series where the $$n$$-th term is $$\frac{n}{n+1}$$, starting from $$n=1$$ to $$n=24$$.

$$
\sum_{n=1}^{24} \frac{n}{n+1}
$$

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**(f) $$5 + 5 + 5 + 5 + 5 + \ldots$$ to $$n$$ terms**

This is a constant series where each term is 5. The general term is simply 5.

$$
\sum_{k=1}^{n} 5
$$

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These sigma notations succinctly represent each of the given series.
Here are the series expressed in sigma notation: **(a)** $$ \sum_{n=1}^{6} 2n $$ **(b)** $$ \sum_{n=1}^{5} n^{3} $$ **(c)** $$ \sum_{n=1}^{7} \left(\frac{1}{2}\right)^n $$ **(d)** $$ \sum_{k=1}^{n} 3 \cdot (-2)^{k-1} $$ **(e)** $$ \sum_{n=1}^{24} \frac{n}{n+1} $$ **(f)** $$ \sum_{k=1}^{n} 5 $$

Write each of the following series in sigma notation:

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