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Naya [18.7K]
3 years ago
15

3 ounces of perfume for $99, 8 ounces for x

Mathematics
1 answer:
marusya05 [52]3 years ago
6 0

Answer:

Step-by-step explanation:

$99÷3=$33 per ounce

8 x $33 = 264

x = $264

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Let a1, a2, a3, ... be a sequence of positive integers in arithmetic progression with common difference
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and since b_1,b_2,b_3,\cdots are in geometric progression,

b_2 = 2b_1

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Recall that

\displaystyle \sum_{k=1}^n 1 = \underbrace{1+1+1+\cdots+1}_{n\,\rm times} = n

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It follows that

a_1 + a_2 + \cdots + a_n = \displaystyle \sum_{k=1}^n (a_1 + 2(k-1)) \\\\ ~~~~~~~~ = a_1 \sum_{k=1}^n 1 + 2 \sum_{k=1}^n (k-1) \\\\ ~~~~~~~~ = a_1 n +  n(n-1)

so the left side is

2(a_1+a_2+\cdots+a_n) = 2c n + 2n(n-1) = 2n^2 + 2(c-1)n

Also recall that

\displaystyle \sum_{k=1}^n ar^{k-1} = \frac{a(1-r^n)}{1-r}

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b_1 + b_2 + \cdots + b_n = \displaystyle \sum_{k=1}^n 2^{k-1}b_1 = c(2^n-1)

Solve for c.

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Now, the numerator increases more slowly than the denominator, since

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This means we only need to check if the claim is true for any n\in\{1,2,3,4\}.

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c = \dfrac{4}{2^2 - 4 - 1} = \dfrac4{-1} = -4 < 0

If n=3, then

c = \dfrac{12}{2^3 - 6 - 1} = 12

If n=4, then

c = \dfrac{24}{2^4 - 8 - 1} = \dfrac{24}7 \not\in\Bbb N

There is only one value for which the claim is true, c=12.

3 0
2 years ago
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