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Westkost [7]
3 years ago
5

How can variables and mayhem expressions be used to show different ways of a pattern?

Mathematics
1 answer:
earnstyle [38]3 years ago
4 0

Answer:

A variable is a letter or symbol, such as a, b, or x, that represents a number. An algebraic expression is a combination of one or more variables; it may include numbers and operation signs. For example, 2 d 5 is an algebraic expression that could represent two times the figure number plus five.

Step-by-step explanation:

<em>Hope</em><em> </em><em>it</em><em> </em><em>helps</em><em> </em><em>u</em><em>.</em><em>.</em><em>pls</em><em> </em><em>mark</em><em> </em><em>me</em><em> </em><em>the</em><em> </em><em>Brainliest</em><em>.</em><em>p</em><em>l</em><em>s</em><em>.</em><em>.</em><em>.</em>

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<img src="https://tex.z-dn.net/?f=%24a%2Ba%20r%2Ba%20r%5E%7B2%7D%2B%5Cldots%20%5Cinfty%3D15%24%24a%5E%7B2%7D%2B%28a%20r%29%5E%7B
riadik2000 [5.3K]

Let

S_n = \displaystyle \sum_{k=0}^n r^k = 1 + r + r^2 + \cdots + r^n

where we assume |r| < 1. Multiplying on both sides by r gives

r S_n = \displaystyle \sum_{k=0}^n r^{k+1} = r + r^2 + r^3 + \cdots + r^{n+1}

and subtracting this from S_n gives

(1 - r) S_n = 1 - r^{n+1} \implies S_n = \dfrac{1 - r^{n+1}}{1 - r}

As n → ∞, the exponential term will converge to 0, and the partial sums S_n will converge to

\displaystyle \lim_{n\to\infty} S_n = \dfrac1{1-r}

Now, we're given

a + ar + ar^2 + \cdots = 15 \implies 1 + r + r^2 + \cdots = \dfrac{15}a

a^2 + a^2r^2 + a^2r^4 + \cdots = 150 \implies 1 + r^2 + r^4 + \cdots = \dfrac{150}{a^2}

We must have |r| < 1 since both sums converge, so

\dfrac{15}a = \dfrac1{1-r}

\dfrac{150}{a^2} = \dfrac1{1-r^2}

Solving for r by substitution, we have

\dfrac{15}a = \dfrac1{1-r} \implies a = 15(1-r)

\dfrac{150}{225(1-r)^2} = \dfrac1{1-r^2}

Recalling the difference of squares identity, we have

\dfrac2{3(1-r)^2} = \dfrac1{(1-r)(1+r)}

We've already confirmed r ≠ 1, so we can simplify this to

\dfrac2{3(1-r)} = \dfrac1{1+r} \implies \dfrac{1-r}{1+r} = \dfrac23 \implies r = \dfrac15

It follows that

\dfrac a{1-r} = \dfrac a{1-\frac15} = 15 \implies a = 12

and so the sum we want is

ar^3 + ar^4 + ar^6 + \cdots = 15 - a - ar - ar^2 = \boxed{\dfrac3{25}}

which doesn't appear to be either of the given answer choices. Are you sure there isn't a typo somewhere?

7 0
2 years ago
Meredith is saving up to buy a digital camera that costs $490. So far, she saved $175. She would like to buy the camera 4 weeks
MatroZZZ [7]

She would have to get 78.75 dollars every week to get the camera.

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3 years ago
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Please Help me solve these problems.
Harman [31]
Which question? The picture is black
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Identify the polygon and classify it as regular or irregular.
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A regular hexagon has sides that are all congruent and angles that all measure 120 degrees. This means the angles of a regular hexagon add up to 720 degrees. ... An irregular hexagon has sides that are not the same measurement and can have points facing inward as well as outward.

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3 years ago
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How to convert 5 1/4 into a decimal
Katen [24]
5 1/4 as a decimal is 5.25

this is because 1/4 is equal to 25/100
5 0
3 years ago
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