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2 years ago

7. A, B, C, and D are positive integers. A,B,C forms an arithmetic sequence while B, C, D forms a

Mathematics

1 answer:

Stella [2.4K]2 years ago

3 0

Complete Question

The positive integers A, B, and C form an arithmetic sequence while the integers B, C, and D form a geometric sequence. If (C/B) = (5/3), what is the smallest possible value of A + B + C + D?

Answer:

Step-by-step explanation:

If A, B, and C form an arithmetic progression

Their arithmetic mean, $B=\dfrac{A+C}{2}$

2B=A+C

C= 2B-A

B, C, D forms a geometric sequence and Common ratio, r=C/B=5/3

The terms in the geometric sequence are:

$B, B(\frac{5}{3} ), B(\frac{5}{3} )^2=B, \frac{5B}{3} , \frac{25B}{9}$

Therefore:

$C=\frac{5B}{3}\\D= \frac{25B}{9}$

So:

$A, B, C, D=A, B, \frac{5B}{3} , \frac{25B}{9}$

From arithmetic sequence

Common difference, $d=B - A = \frac{5B}{3} - B$

$2B -\frac{5B}{3}=A$

$(2 -\frac{5}{3})B=A\\(\frac{1}{3})B=A\\A=\frac{B}{3}\\$

$A, B,C, D =\frac{B}{3},\;B, \;\frac{5B}{3},\;\frac{25B}{9}$

These all have to be positive integers so B must be a multiple of 9, The smallest values are if B is 9

A,B,C,D=3,9,15,25

So the smallest possible value for:

A+B+C+D = 3+9+15+25 = 52

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where
$f(t)$ is the final amount after $t$ years
$a$ is the initial amount
$b$ is the decay or grow rate rate in decimal form
$t$ is the time in years

Expression A
$f(t)=624(0.95)^{4t}$
Since the base (0.95) is less than one, we have a decay rate here.
Now to find the rate $b$ , we are going to use the formula: $b=|1-base|$ *100%
$b=|1-0.95|$ *100%
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Now, to find the initial value of the function, we are going to evaluate the function at $t=0$
$f(t)=624(0.95)^{4t}$
$f(0)=624(0.95)^{0t}$
$f(0)=624(0.95)^{0}$
$f(0)=624(1)$
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Expression B
$f(t)=725(1.12)^{3t}$
Since the base (1.12) is greater than 1, we have a growth rate here.
To find the rate, we are going to use the same equation as before:
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We can conclude that expression B grows at a rate of 12% every 4 months.

Just like before, to find the initial value of the expression, we are going to evaluate it at $t=0$
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$f(0)=725(1.12)^{0t}$
$f(0)=725(1.12)^{0}$
$f(0)=725(1)$
$f(0)=725$
The initial value of expression B is 725.

We can conclude that you should select the statements:
- Expression A decays at a rate of 5% every three months, while expression B grows at a rate of 12% every fourth months.

- Expression A has an initial value of 624, while expression B has an initial value of 725.

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