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cestrela7 [59]
3 years ago
15

Find the unknown value in the pair of equivalent fractions. 1/8 = ?/4?

Mathematics
1 answer:
mars1129 [50]3 years ago
4 0

Answer:

The unknown value is 1/2

Step-by-step explanation:

Let

x ----> the unknown value

we have

\frac{1}{8}=\frac{x}{4}

Solve for x

Multiply in cross

(1)(4)=(x)(8)\\4=8x

Divide by 8 both sides

x=\frac{1}{2}

therefore

The unknown value is 1/2

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I need help with this problem from the calculus portion on my ACT prep guide
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Given a series, the ratio test implies finding the following limit:

\lim _{n\to\infty}\lvert\frac{a_{n+1}}{a_n}\rvert=r

If r<1 then the series converges, if r>1 the series diverges and if r=1 the test is inconclusive and we can't assure if the series converges or diverges. So let's see the terms in this limit:

\begin{gathered} a_n=\frac{2^n}{n5^{n+1}} \\ a_{n+1}=\frac{2^{n+1}}{(n+1)5^{n+2}} \end{gathered}

Then the limit is:

\lim _{n\to\infty}\lvert\frac{a_{n+1}}{a_n}\rvert=\lim _{n\to\infty}\lvert\frac{n5^{n+1}}{2^n}\cdot\frac{2^{n+1}}{\mleft(n+1\mright)5^{n+2}}\rvert=\lim _{n\to\infty}\lvert\frac{2^{n+1}}{2^n}\cdot\frac{n}{n+1}\cdot\frac{5^{n+1}}{5^{n+2}}\rvert

We can simplify the expressions inside the absolute value:

\begin{gathered} \lim _{n\to\infty}\lvert\frac{2^{n+1}}{2^n}\cdot\frac{n}{n+1}\cdot\frac{5^{n+1}}{5^{n+2}}\rvert=\lim _{n\to\infty}\lvert\frac{2^n\cdot2}{2^n}\cdot\frac{n}{n+1}\cdot\frac{5^n\cdot5}{5^n\cdot5\cdot5}\rvert \\ \lim _{n\to\infty}\lvert\frac{2^n\cdot2}{2^n}\cdot\frac{n}{n+1}\cdot\frac{5^n\cdot5}{5^n\cdot5\cdot5}\rvert=\lim _{n\to\infty}\lvert2\cdot\frac{n}{n+1}\cdot\frac{1}{5}\rvert \\ \lim _{n\to\infty}\lvert2\cdot\frac{n}{n+1}\cdot\frac{1}{5}\rvert=\lim _{n\to\infty}\lvert\frac{2}{5}\cdot\frac{n}{n+1}\rvert \end{gathered}

Since none of the terms inside the absolute value can be negative we can write this with out it:

\lim _{n\to\infty}\lvert\frac{2}{5}\cdot\frac{n}{n+1}\rvert=\lim _{n\to\infty}\frac{2}{5}\cdot\frac{n}{n+1}

Now let's re-writte n/(n+1):

\frac{n}{n+1}=\frac{n}{n\cdot(1+\frac{1}{n})}=\frac{1}{1+\frac{1}{n}}

Then the limit we have to find is:

\lim _{n\to\infty}\frac{2}{5}\cdot\frac{n}{n+1}=\lim _{n\to\infty}\frac{2}{5}\cdot\frac{1}{1+\frac{1}{n}}

Note that the limit of 1/n when n tends to infinite is 0 so we get:

\lim _{n\to\infty}\frac{2}{5}\cdot\frac{1}{1+\frac{1}{n}}=\frac{2}{5}\cdot\frac{1}{1+0}=\frac{2}{5}=0.4

So from the test ratio r=0.4 and the series converges. Then the answer is the second option.

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