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2 years ago

6. If four numbers are in proportion, then the terms equals the product of

1 answer:

3 0

Answer: If four numbers are proportional, a : b = c : d, then the product of the extremes is equal to the product of the means. ad = bc

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Three numbers form a geometric progression. If the second term is increased by 2, then the progression will become arithmetic an

Margaret [11]

The three numbers are 4 , 8 , 16

Step-by-step explanation:

Let us revise how to find the nth term of the geometric and the arithmetic progressions

The nth term of the geometric sequence is $a_{n}=ar^{n-1}$ , where a is the first term and r is the common ratio between the consecutive terms
The nth term of the arithmetic sequence is $a_{n}=a+(n-1)d$ , where a is the first term and d in the common difference between the consecutive terms

∵ Three numbers form a geometric progression

- Assume that the first number is a and the common ratio is r

and n = 1 , 2 , 3

∴ The three terms are a , ar and ar²

∵ The second term is increased by 2

∴ The three terms are a , ar + 2 , ar²

∵ The three terms formed an arithmetic progression

- The common difference between each two consecutive terms is d

∴ d = ar + 2 - a and d = ar² - (ar + 2)

- Equate the right hand sides of d

∴ ar + 2 - a = ar² - ar - 2

- Add 2 to both sides

∴ ar - a + 4 = ar² - ar

- Subtract ar from both sides

∴ -a + 4 = ar² - 2ar

- Add a to both sides

∴ 4 = ar² - 2ar + a

- Take a as a common factor in the right hand side

∴ 4 = a(r² - 2r + 1)

∵ r² - 2r + 1 = (r - 1)²

∴ 4 = a(r - 1)²

- Divide both sides by (r - 1)²

∴ $a=\frac{4}{(r-1)^{2}}$ ⇒ (1)

∵ The last term is increased by 9

∴ The three terms are a , ar + 2 , ar² + 9

∵ The three terms formed an geometric progression

- Find the common ratio between each two consecutive terms

∵ The common ratio = $\frac{ar+2}{a}$

∵ The common ratio = $\frac{ar^{2}+9}{ar+2}$

- Equate the right hand sides of the common ratio

∴ $\frac{ar+2}{a}=\frac{ar^{2}+9}{ar+2}$

- By using cross multiplication

∴ a(ar² + 9) = (ar + 2)²

- Simplify the two sides

∴ a²r² + 9a = a²r² + 4ar + 4

- Subtract a²r² from both sides

∴ 9a = 4ar + 4

- Subtract 4ar from both sides

∴ 9a - 4ar = 4

- Take a as a common factor from both sides

∴ a(9 - 4r) = 4

- Divide both sides by (9 - 4r)

∴ $a=\frac{4}{(9-4r)}$ ⇒ (2)

Equate the right hand sides of (1) and (2)

∴ $\frac{4}{(r-1)^{2}}$ = $\frac{4}{(9-4r)}$

- By using cross multiplication

∴ 4(r - 1)² = 4(9 - 4r)

- Divide both sides by 4

∴ (r - 1)² = 9 - 4r

- Solve the bracket of the left hand side

∴ r² - 2r + 1 = 9 - 4r

- Add 4r to both sides

∴ r² + 2r + 1 = 9

- Subtract 9 from both sides

∴ r² + 2r - 8 = 0

- Factorize it into 2 factors

∴ (r - 2)(r + 4) = 0

- Equate each factor by 0

∵ r - 2 = 0

- Add 2 to both sides

∴ r = 2

∵ r + 4 = 0

- Subtract 4 from both sides

∴ r = -4 ⇒ rejected

Substitute the value of r in equation (2) to find a

∵ $a=\frac{4}{9-4(2)}=\frac{4}{9-8}=\frac{4}{1}$

∴ a = 4

∵ The three numbers are a , ar , ar²

∵ a = 4 and r = 2

∴ The numbers are 4 , 4(2) , 4(2)²

∴ The numbers are 4 , 8 , 16

Lets check the answer

4 , 8 + 2 , 16 ⇒ 4 , 10 , 16 formed an arithmetic progression with common difference 6

4 , 10 , 16 + 9 ⇒ 4 , 10 , 25 formed a geometric progression with common ratio 2.5

The three numbers are 4 , 8 , 16

Learn more:

You can learn more about the progressions in brainly.com/question/1522572

#LearnwithBrainly

5 0

3 years ago

Complete the pattern, 1.1, 2.2, 3-3. What is the rule?

spin [16.1K]

You mean 3.3? If that is the case then:

an=1.1+1(n-2)

or an=an-1+1
a1= 1.1

5 0

3 years ago

Factor the expression x^4 -1 using the polynomial identify x^2-y^2)=(x-y)(x+y)

Orlov [11]

Answer:

(x - 1)(x + 1)(x² + 1)

Step-by-step explanation:

Step 1: Factor Difference of Squares

(x² - 1)(x² + 1)

Step 2: Factor Difference of Squares for non-complex binomial root

(x - 1)(x + 1)(x² + 1)

And we have our answer!

3 0

3 years ago

Explain how to use the pattern in the table to rewrite a division problem involving fractions as a multiplication problem

Irina-Kira [14]

Reverse the fraction

7 0

3 years ago

Can someone help me with this?

Anna35 [415]

Answer:

Let 'a' be the first term, 'r' be the common ratio and 'n' be the number of terms

Series = 2+6+18.......= 2+2•3¹+ 2•3².......= 728

Now,

$Sum = \frac{a( {r}^{n} - 1) }{(r - 1)} \\$

So,

$\frac{a( {r}^{n} - 1)}{(r - 1)} = 728 \\ \frac{2( {3}^{n} - 1) }{(3 - 1)} = 728 \\ \frac{2( {3}^{n} - 1) }{2} = 728 \\ {3}^{n} - 1 = 728 \\ {3}^{n}=728+1\\ {3}^{n} = 729 \\ {3}^{n} = {3}^{6} \\ \boxed{ n = 6}$

Therefore, number of terms is 6

6 is the right answer.

3 0

3 years ago

6. If four numbers are in proportion, then the terms equals the product of​

6. If four numbers are in proportion, then the terms equals the product of