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2 years ago

11

If numerator is 2 less than the denominator of a rational number and when 1 is subtracted from numerator and denominator both, t

he resulting new rational number in its simplest form is 1/2 . Find the original rational number.

1 answer:

LUCKY_DIMON [66]2 years ago

6 0

Answer:

3/5

Step-by-step explanation:

1/2=2/4

2+1=3

1+4=5

3/5

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If S_1=1,S_2=8 and S_n=S_n-1+2S_n-2 whenever n≥2. Show that S_n=3⋅2n−1+2(−1)n for all n≥1.

Snezhnost [94]

You can try to show this by induction:

• According to the given closed form, we have $S_1=3\times2^{1-1}+2(-1)^1=3-2=1$ , which agrees with the initial value <em>S</em>₁ = 1.

• Assume the closed form is correct for all <em>n</em> up to <em>n</em> = <em>k</em>. In particular, we assume

$S_{k-1}=3\times2^{(k-1)-1}+2(-1)^{k-1}=3\times2^{k-2}+2(-1)^{k-1}$

and

$S_k=3\times2^{k-1}+2(-1)^k$

We want to then use this assumption to show the closed form is correct for <em>n</em> = <em>k</em> + 1, or

$S_{k+1}=3\times2^{(k+1)-1}+2(-1)^{k+1}=3\times2^k+2(-1)^{k+1}$

From the given recurrence, we know

$S_{k+1}=S_k+2S_{k-1}$

so that

$S_{k+1}=3\times2^{k-1}+2(-1)^k + 2\left(3\times2^{k-2}+2(-1)^{k-1}\right)$

$S_{k+1}=3\times2^{k-1}+2(-1)^k + 3\times2^{k-1}+4(-1)^{k-1}$

$S_{k+1}=2\times3\times2^{k-1}+(-1)^k\left(2+4(-1)^{-1}\right)$

$S_{k+1}=3\times2^k-2(-1)^k$

$S_{k+1}=3\times2^k+2(-1)(-1)^k$

$\boxed{S_{k+1}=3\times2^k+2(-1)^{k+1}}$

which is what we needed. QED

6 0

2 years ago

The length of a rectangle is seven more than double the width. If the perimeter is 22 inches find the dimensions

lubasha [3.4K]

Answer:

width = 1 1/3 inches

length = 9 2/3 inches

Step-by-step explanation:

let 'w' = width

let '7+2w' = length

P = 2l + 2w

22 = 2w + 2(7+2w)

22 = 2w + 14 + 4w

8 = 6w

w = 4/3 or 1 1/3

substitute 1 1/3, or 4/3, to find length:

l = 7 + 2(4/3)

l = 7 + 8/3 or 9 2/3

7 0

3 years ago

The graph of y = f(x) is shown below. What are all of the real solutions of f(x) = 0?

jasenka [17]

Answer:

-8 and -2

Step-by-step explanation:

f(x) basically means y and at the y value 0 the x values are -8 and -2 (also known as the roots of the function)

5 0

2 years ago

Which expressions are equivalent to the one below? Check all that apply. <br><br> 9^x

AveGali [126]

Answer:

A) $3^{x}*3^{x}$

B) $3^{2x}$

C) $(3 * 3)^{x}$

Step-by-step explanation:

Given the exponential expression, $9^{x}$ :

A) $3^{x}*3^{x}$ is equivalent to $9^{x}$ due to the Product Rule of exponents: $a^{m} a^{n} = a^{m + n}$ .

$3^{x}*3^{x} = 3^{x+x} = 3^{2x}$

Next, apply the Power-to-Power Rule of exponents: $a^{mn} = (a^{m} )^{n}$ .

$3^{x}*3^{x} = 3^{x+x} = 3^{2x} = (3^{2})^{x} = 9^{x}$

B) $3^{2x}$ is equivalent to $9^{x}$ due to the Power-to-Power Rule of exponents: $a^{mn} = (a^{m} )^{n}$ .

$3^{2x} = (3^{2})^{x} = (9)^{x} = 9^{x}$ .

C) $(3 * 3)^{x}$ is equivalent to $9^{x}$ due to the Product-to-Power Rule of exponents: $(ab)^{m} = a^{m} b^{m}$ .

$(3 * 3)^{x} = (9)^{x} = 9^{x}$

5 0

2 years ago

Help me please<br> Idk what to do

Ber [7]

Answer:

x=36

Step-by-step explanation:

First rewrite equation then multiply each side by 36. 9x-36=4x+144.

then move the variable 9x-4x+36=144. Next subtract 9x and 4x ...5x=144+36.... 5x=180. Lastly you divide 180÷5 this your answer x=36

4 0

2 years ago