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Vlad [161]
3 years ago
9

Prove that an = 4^n + 2(-1)^nis the solution to

Mathematics
1 answer:
olga nikolaevna [1]3 years ago
4 0

Answer:

See proof below

Step-by-step explanation:

We have to verify that if we substitute a_n=4^n+2(-1)^n in the equation a_n=3a_{n-1}+4a_{n-2} the equality is true.

Let's substitute first in the right hand side:

3a_{n-1}+4a_{n-2}=3(4^{n-1}+2(-1)^{n-1})+4(4^{n-2}+2(-1)^{n-2})

Now we use the distributive laws. Also, note that (-1)^{n-1}=\frac{1}{-1}(-1)^n=(-1)(-1)^{n} (this also works when the power is n-2).

=3(4^{n-1})+6(-1)^{n-1}+4(4^{n-2})+8(-1)^{n-2}

=3(4^{n-1})+(-1)(6)(-1)^{n}+4^{n-1}+(-1)^2(8)(-1)^{n}

=4(4^{n-1})-6(-1)^{n}+8(-1)^{n}=4^n+2(-1)^n=a_n

then the sequence solves the recurrence relation.

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Am I correct? Just check please
creativ13 [48]

Answer:

Yes..You are

Step-by-step explanation:

|Wow! Nice Job! Even though Absolute value is EXTREMELy EASY you probably aced the test!

6 0
2 years ago
(a) How many integers from 1 through 1,000 are multiples of 2 or multiples of 9?
DENIUS [597]

Answer:

(a)There are 500 integers from 1 through 1,000 are multiples of 2.

There are 111 integers from 1 through 1,000 are multiples of 9.

(b)0.611

Step-by-step explanation:

Given the set of Integers from 1 through 1000

The least multiple of 2 is 2 and the highest multiple of 2 in the interval is 1000.

To determine the number of terms, we use the formula for the nth term of an Arithmetic Progression, since 2,4,6,... is an Arithmetic Progression,

where first term, a =2, common difference, d =2

Nth term of an A.P,

U_n= a+(n-1)d\\1000=2+2(n-1)\\1000=2+2n-2\\1000=2n\\n=500

  • There are 500 integers from 1 through 1,000 are multiples of 2.

Similarly,

Given the set of Integers from 1 through 1000

The least multiple of 9 is 9 and the highest multiple of 9 in the interval is 999.

To determine the number of terms, we use the formula for the nth term of an Arithmetic Progression, since 9,18,27,... is an Arithmetic Progression,

where first term, a =9, common difference, d =9

Nth term of an A.P,

U_n= a+(n-1)d\\999=9+9(n-1)\\999=9+9n-9\\999=9n\\n=111

  • There are 111 integers from 1 through 1,000 are multiples of 9.

(b)Probability that an integer selected at random is a multiple of 2 or a multiple of 9.

There are a total of 1000 numbers between from 1 through 1,000

n(S)=1000

n(Multiples of 2)=500

n(Multiples of 9)=111

Therefore:

P(\text{Multiples of 9}) \:OR\: P(\text{Multiples of 2})=\frac{111}{1000} + \frac{500}{1000} \\=\frac{611}{1000} \\=0.611

7 0
3 years ago
The altitude of an equilateral triangle is 15. What is the length of each side?
Bogdan [553]
An equilateral triangle is a triangle where all sides are of equal lengths. So, the angles are of equal values as well which is 60. We use the angle and the height of the triangle to determine the side length. We do as follows:

tan (60) = 15 / base/2
base = 10√3 = side length
6 0
2 years ago
Would appreciate the help 24 points
andre [41]

Answer:

13) Rectangle

14) parallelogram

15) co ordinates areD (0,a) andE (c,b)

16) co ordinates are D(0,b) and E(a,0)

17)(-1,1) (4,1) (-1,4)

18)(0,0) (a,0) (a,a) (0,a)

Step-by-step explanation:

<h3>13)</h3><h3>Rectangle</h3>

The opposite sides of a rectangle are parallel and equal

left height = right height

(by using distance formula)

(0,3),(2,-2) = (5,5),(7,0)

√(2-0)²+(-2-3)² = √(7-5)²+(0-5)²

√(2)²+(-5)² = √(2)²+(-5)²

√4+25 = √(2)²+(-5)²

√29 = √29 hence equal

<h3>14)</h3><h3>parallelogram</h3>

The opposite sides of a rectangle are not equal

left height = right height

(by using distance formula)

(0,0),(3,-4) = (3,4),(7,0)

√(3-0)²+(-4-0)² = √(7-3)²+(0-4)²

√(3)²+(-4)² = √(5)²+(-4)²

√9+16 = √(25)+(16)

√(25) ≠√41

5 ≠ √41

<h3>15)</h3><h3>co ordinates are D(0,a) and E(c,b)</h3><h3>16)</h3><h3>co ordinates are D(0,b) and E(a,0) </h3>

since rhombus have equal opposite sides

<h3>17)</h3><h3>(-1,1) (4,1) (-1,4)</h3><h3>18)</h3><h3>(0,0) (a,0) (a,a) (0,a)</h3><h3 />

6 0
3 years ago
In a system of linear equations in two​ variables, if the graphs of the equations are the​ same, the equations are (blank) equat
makkiz [27]

Answer:

Please check the explanation.

Step-by-step explanation:

We know that when a consistent system has infinite solutions, then the graphs of the equations are exactly the same. In other words, these equations are called dependent equations.

All points of dependent equations share the same slope and same y-intercept.

For example,

6x-2y = 18

9x-3y=27

represent the dependent equations.

Writing both equations in slope-intercept form

y=mx+c

where m is the slope and c is the y-intercept

Now

6x-2y=18

2y = 6x-18

Divide both sides by 2

y = 3x - 9

Thus, the slope = 3 and y-intercept = b = -9

now

9x-3y=27

3y = 9x-27

Divide both sides by 3

y = 3x - 9

Thus, the slope = 3 and y-intercept = b = -9

Therefore, both equations have the same slope and y-intercept. Their graphs are the same. Hence, they are called dependent equations.

5 0
2 years ago
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