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padilas [110]
3 years ago
13

Write and solve the proportion.

Mathematics
2 answers:
mart [117]3 years ago
6 0

Answer:

5 gallons of gas cost about $15.42

Nonamiya [84]3 years ago
5 0

Answer: I believe it would be 15.42

Step-by-step explanation:

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Let a1, a2, a3, ... be a sequence of positive integers in arithmetic progression with common difference
Bezzdna [24]

Since a_1,a_2,a_3,\cdots are in arithmetic progression,

a_2 = a_1 + 2

a_3 = a_2 + 2 = a_1 + 2\cdot2

a_4 = a_3+2 = a_1+3\cdot2

\cdots \implies a_n = a_1 + 2(n-1)

and since b_1,b_2,b_3,\cdots are in geometric progression,

b_2 = 2b_1

b_3=2b_2 = 2^2 b_1

b_4=2b_3=2^3b_1

\cdots\implies b_n=2^{n-1}b_1

Recall that

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

\displaystyle \sum_{k=1}^n k = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}2

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}

so that the right side is

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

Solve for c.

2n^2 + 2(c-1)n = c(2^n-1) \implies c = \dfrac{2n^2 - 2n}{2^n - 2n - 1} = \dfrac{2n(n-1)}{2^n - 2n - 1}

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

\dfrac{d}{dn}(2n(n-1)) = 4n - 2

\dfrac{d}{dn} (2^n-2n-1) = \ln(2)\cdot2^n - 2

and for n\ge5,

2^n > \dfrac4{\ln(2)} n \implies \ln(2)\cdot2^n - 2 > 4n - 2

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

n=1 doesn't work, since that makes c=0.

If n=2, then

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
Find the difference between 25.6 and 14.07
WINSTONCH [101]

Answer:

25.6-14.07= 11.53

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
find the value of K so that the line passing thru the points (K,-1) and (-2,-8) is parrallel to the equation 3x+y= -7​
fgiga [73]

Answer:

K=-13/3

Step-by-step explanation:

8 0
2 years ago
John needs 13 bottles of water from the store. John can only carry 3 at a time. What's the minimum number of trips John needs to
artcher [175]
13/3 = 4 1/3...so the minimum would be 4 1/2
6 0
3 years ago
Read 2 more answers
Anyone smart in geometry?
mr_godi [17]

Answer:

Jml for number one because if you flip the top triangle over you have two like triangles just not the same size but angles are the same

Number 25 edc because of the same reason

Step-by-step explanation:

3 0
2 years ago
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