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In-s [12.5K]
2 years ago
14

2%7D%2B%5Cleft%28a%20r%5E%7B2%7D%5Cright%29%5E%7B2%7D%2B%5Cldots%20%5Cinfty%3D150%24.%20Find%20%24a%20r%5E%7B3%7D%2Ba%20r%5E%7B4%7D%2Ba%20r%5E%7B6%7D%2B%5Cldots%20%5Cinfty%24" id="TexFormula1" title="$a+a r+a r^{2}+\ldots \infty=15$$a^{2}+(a r)^{2}+\left(a r^{2}\right)^{2}+\ldots \infty=150$. Find $a r^{3}+a r^{4}+a r^{6}+\ldots \infty$" alt="$a+a r+a r^{2}+\ldots \infty=15$$a^{2}+(a r)^{2}+\left(a r^{2}\right)^{2}+\ldots \infty=150$. Find $a r^{3}+a r^{4}+a r^{6}+\ldots \infty$" align="absmiddle" class="latex-formula">
Options:
(a) $\frac{1}{2}$\\(b) $\frac{2}{5}$
Mathematics
1 answer:
riadik2000 [5.3K]2 years ago
7 0

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?

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Answer:

                            \large\boxed{\large\boxed{\sqrt{3}

Explanation:

You are comparing irrational numbers.

By inspection, i.e. at first sight you can only compare \sqrt{3} \text{ }and\text{ } 2\sqrt{3} because they have the same radicand.

You can order: \sqrt{3}

You can introduce the 2 inside the radical by squaring it:

       2\sqrt{3}=\sqrt{2^2\times3}=\sqrt{12}

Since 5 is between 3 and 12, you can order:

  • \sqrt{3}

Which is:

  • \sqrt{3}

You must know that π ≈ 3.14.

5 is less than 9 and the square root of 9 is 3; hence, \sqrt{5} and \sqrt{5}

Now you must determine whether π is less than or greater than \sqrt{12}

Using a calculator or probing numbers between 3 and 4 you get \sqrt{12} \approx3.46

Hence, the complete order is:

  •            \sqrt{3}
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Question 8<br> Calculate the difference (4x2 + x - 5) – (5x2 - 1).
Rufina [12.5K]

Answer:

x - 6

Step-by-step explanation:

(4x2 + x - 5) - (5x2 - 1)

According to the order of operations, parenthesis come first so we will solve what is in the parenthesis first.

Then we do the multiplication because it comes before addition and subtraction.

4x2 = 8

Now we have 8 + x - 5

Combine like terms and 8-5=3 so we have 3 + x

Now for the second one.

5x2=10

10-1=9

Subtracting this from the first expression we have:

3+x-9 =

x - 6

7 0
2 years ago
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