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1 year ago

Find the x if the sequence 3, x, 4x/3 is (a)arithmetic and (b)geometric

Mathematics

1 answer:

jolli1 [7]1 year ago

6 0

(a) When the sequence is arithmetic, sequential terms have a common difference.

... x - 3 = (4x/3) - x . . . . differences of sequential terms are equal

... (2/3)x = 3 . . . . . . . add 3-(1/3)x

... x = 9/2 . . . . . . . . . multiply by 3/2

(The arithmetic sequence is 3, 4.5, 6. The common difference is 3/2.)

(b) When the sequence is geometric, sequential terms have a common ratio.

... x/3 = (4x/3)/x . . . . . ratios of sequential terms are equal

... x^2 = 4x . . . . . multiply by 3x

... x = 4 . . . . . . . . divide by x. (the "solution" x=0 is extraneous)

(The geometric sequence is 3, 4, 16/3. The common ratio is 4/3.)

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Show that the Fibonacci numbers satisfy the recurrence relation fn = 5fn−4 + 3fn−5 for n = 5, 6, 7, . . . , together with the in

Sonja [21]

Answer with step-by-step explanation:

We are given that the recurrence relation

$f_n=5f_{n-4}+3f_{n-5}$

for n=5,6,7,..

Initial condition

$f_0=0,f_1=1,f_2=1,f_3=2,f_4=3$

We have to show that Fibonacci numbers satisfies the recurrence relation.

The recurrence relation of Fibonacci numbers

$f_n=f_{n-1}+f_{n-2}$ , $f_0=0,f_1=1$

Apply this

$f_n=(f_{n-2}+f_{n-3})+f_{n-2}=2f_{n-2}+f_{n-3}$

$f_n=2(f_{n-3}+f_{n-4})+f_{n-3}=3f_{n-3}+2f_{n-4}$

$f_n=3(f_{n-4}+f_{n-5})+2f_{n-4}=5f_{n-4}+3f_{n-5}$

Substitute n=2

$f_2=f_1+f_0=1+0=1$

$f_3=f_2+f_1=1+1=2$

$f_4=f_3+f_2=2+1=3$

Hence, the Fibonacci numbers satisfied the given recurrence relation .

Now, we have to show that $f_{5n}$ is divisible by 5 for n=1,2,3,..

Now replace n by 5n

$f_{5n}=5f_{5n-4}+3f_{5n-5}$

Apply induction

Substitute n=1

$f_5=5f_1+3f_0=5+0=5$

It is true for n=1

Suppose it is true for n=k

$f_{5k}=5f_{5k-4}+3f_{5k-5}$ is divisible 5

Let $f_{5k}=5q$

Now, we shall prove that for n=k+1 is true

$f_{5k+5}=5f_{5k+5-4}+3f_{5k+5-5}=5f_{5k+1}+3f_{5k}=5f_{5k+1}+3(5q)$

$f_{5k+5}=5(f_{5k+1}+3q)$

It is multiple of 5 .Therefore, it is divisible by 5.

It is true for n=k+1

Hence, the $f_{5n}$ is divisible by 5 for n=1,2,3,..

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1 year ago

Mr. Henry spent $15 on a new hat this week. He learned that the same hat available on

Arlecino [84]

Answer: The answer is 25 percent

Step-by-step explanation:

A math man doesn't reveal his explanation, also you should watch jojo bizzare adventures, it's a really good anime

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2 years ago

Clue is a board game in which you must deduce three details surrounding a murder. In the original game of Clue, the guilty perso

Step2247 [10]

Answer:

the probability is 0.1143

Step-by-step explanation:

here are the details from the question you asked

number of people = 6

nuber of weapons = 6

number of rooms = 9

after narrowing it down to

3 people,3 weapons, and 6 rooms

= ⁶C₃ * ⁶C₃ *⁹C₆ / ²¹C₁₂

= 20 * 20 * 84 / 293930

= 33600/293930

= 0.1143

<u>The probability of making a random guess of the person who is guilty, weapon and location from this choices that have been narrowed down and the guess being correct is 0.1143</u>

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1 year ago

If on a scale drawing 15 feet are represented by 10 inches then a scale of 1/10 inch represents how many feet

GenaCL600 [577]

A scale of $\frac{1}{10}$ inches will represent $\frac{3}{20}$ feet.

Step-by-step explanation:

According to scale drawing;

15 feet = 10 inches

1 inch = $\frac{15}{10}=\frac{3}{2}\ feet$

A scale of $\frac{1}{10}$ inch will represent;

Multiplying both sides by

$\frac{1}{10}*1=\frac{3}{2}*\frac{1}{10}\\\\\frac{1}{10}\ inches = \frac{3}{20}\ feet$

A scale of $\frac{1}{10}$ inches will represent $\frac{3}{20}$ feet.

Keywords: fraction, multiplication

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1 year ago

A business purchased for $650,000 in 1994 is sold in 1997 for $850,000. What is the annual rate of return for this investment?

Schach [20]

The annual rat of return for this investment would be
850000=650000*(1+(r/1))^(1*3)

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2 years ago