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mash [69]
3 years ago
8

Find the sum of the first 25 terms in this geometric series: 8 + 6 + 4.5...

Mathematics
1 answer:
Ksivusya [100]3 years ago
3 0

Step-by-step explanation:

Given the geometric sequence

8 + 6 + 4.5...

A geometric sequence has a constant ratio and is defined by

a_n=a_1\cdot r^{n-1}

\mathrm{Compute\:the\:ratios\:of\:all\:the\:adjacent\:terms}:\quad \:r=\frac{a_{n+1}}{a_n}

\frac{6}{8}=\frac{3}{4},\:\quad \frac{4.5}{6}=\frac{3}{4}

\mathrm{The\:ratio\:of\:all\:the\:adjacent\:terms\:is\:the\:same\:and\:equal\:to}

r=\frac{3}{4}

\mathrm{The\:first\:element\:of\:the\:sequence\:is}

a_1=8

\mathrm{Therefore,\:the\:}n\mathrm{th\:term\:is\:computed\:by}\:

a_n=8\left(\frac{3}{4}\right)^{n-1}

\mathrm{Geometric\:sequence\:sum\:formula:}

a_1\frac{1-r^n}{1-r}

\mathrm{Plug\:in\:the\:values:}

n=25,\:\spacea_1=8,\:\spacer=\frac{3}{4}

=8\cdot \frac{1-\left(\frac{3}{4}\right)^{25}}{1-\frac{3}{4}}

\mathrm{Multiply\:fractions}:\quad \:a\cdot \frac{b}{c}=\frac{a\:\cdot \:b}{c}

=\frac{\left(1-\left(\frac{3}{4}\right)^{25}\right)\cdot \:8}{1-\frac{3}{4}}

=\frac{8\left(-\left(\frac{3}{4}\right)^{25}+1\right)}{\frac{1}{4}}

\mathrm{Apply\:exponent\:rule}:\quad \left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}

=\frac{8\left(-\frac{3^{25}}{4^{25}}+1\right)}{\frac{1}{4}}

\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a}{\frac{b}{c}}=\frac{a\cdot \:c}{b}

=\frac{\left(1-\frac{3^{25}}{4^{25}}\right)\cdot \:8\cdot \:4}{1}

\mathrm{Multiply\:the\:numbers:}\:8\cdot \:4=32

=\frac{32\left(-\frac{3^{25}}{4^{25}}+1\right)}{1}

=\frac{32\cdot \frac{4^{25}-3^{25}}{4^{25}}}{1}               ∵ \mathrm{Join}\:1-\frac{3^{25}}{4^{25}}:\quad \frac{4^{25}-3^{25}}{4^{25}}

=32\cdot \frac{4^{25}-3^{25}}{4^{25}}

=\frac{\left(4^{25}-3^{25}\right)\cdot \:32}{4^{25}}

=\frac{2^5\left(4^{25}-3^{25}\right)}{2^{50}}        ∵ \mathrm{Factor}\:32:\ 2^5,  \mathrm{Factor}\:4^{25}:\ 2^{50}

so

=\frac{4^{25}-3^{25}}{2^{45}}        ∵ \mathrm{Cancel\:}\frac{\left(4^{25}-3^{25}\right)\cdot \:2^5}{2^{50}}:\quad \frac{4^{25}-3^{25}}{2^{45}}

\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a\pm \:b}{c}=\frac{a}{c}\pm \frac{b}{c}

=\frac{4^{25}}{2^{45}}-\frac{3^{25}}{2^{45}}      

=32-\frac{3^{25}}{2^{45}}            ∵  \frac{4^{25}}{2^{45}}=32

=32-0.024        ∵  \frac{3^{25}}{2^{45}}=0.024

=31.98            

Therefore, the sum of the first 25 terms in this geometric series: 31.98

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Answer:

(34, 48)

Step-by-step explanation:

According to the Empirical Rule, 95% of normally distributed data lie within two standard deviations of the mean.  That, in turn, means 95% of the data in this problem lie within 2(3.5 min), or 7 min, of the mean:

41 - 7 < mean < 41 + 7, or

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Parallelogram ABCD is dilated to form parallelogram EFGH. Side BC is proportional to side CD.
egoroff_w [7]
Side FG is proportional to side BC. When showing that two figures are similar, they should always be typed in the order of the points they correspond to. In this instance, ABCD ~ EFGH, meaning that AB ~ EF, BC ~ FG, CD ~ GH, and AD ~ EH. Let's use a simpler example, with similar triangles MNO and XYZ.

MNO ~ XYZ   Since the two triangles are written in this specific order, each side should be similar to the side in the same order on the other triangle.
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3 years ago
Gary made 5 baskets than Susan this season bill made 3 less then twice as many baskets as Susan. Together, all three made 42 bas
Nezavi [6.7K]

Answer:Gary made 15 baskets.

Susan made 10 baskets.

Bill made 17 baskets.

Step-by-step explanation:

Let x represent the number of baskets that Gary made.

Let y represent the number of baskets that Susan made.

Let z represent the number of baskets that Bill made.

Gary made 5 baskets than Susan this season. This means that

x = y + 5

Bill made 3 less than twice as many baskets as Susan. This means that

z = 2y - 3

Together, all three made 42 baskets. This means that

x + y + z = 42 - - - - - - - - - - 1

Substituting x = y + 5 and z = 2y - 3 into equation 1, it becomes

y + 5 + y + 2y - 3 = 42

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3 years ago
Whats the Axis of Symmetry? willing to give 15 points !!
nydimaria [60]

The axis of symmetry of a parabola in the form

y=ax^2+bx+c

is a vertical line

x=k

where k is the x coordinate of the vertex of the parabola.

The x coordinate of the vertex of the parabola is given by

-\dfrac{b}{2a}

which in your case is

-\dfrac{-6}{2\cdot \frac{1}{4}}=\dfrac{6}{\frac{1}{2}}=6\cdot 2=12

So, the axis of symmetry is the line x=12.

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GREYUIT [131]

Answer:

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Step-by-step explanation:

A=1/2(l*w)

A=1/2(5*2)

A=1/2(10)

A=5

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3 years ago
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