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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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Need help with another problem
Verdich [7]
Find the midpoint:

m= x1+x2/2; y1+y2/2

m= 9+-1/2; 8+-2/2

m= 8/2; 6/2

m= (4,3)

(4,3) is your answer.

I hope this helps!
~kaikers
4 0
3 years ago
If a₁ = 4 and an = 5an-1 then find the value of a5.
igor_vitrenko [27]

The value of a_{5} is 2500, when a_{1}=4 and 5a_{n-1}.

Given that, a_{1}=4 and 5a_{n-1}.

We need to find the value of a_{5}.

<h3>What is an arithmetic sequence?</h3>

An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

Now, to find the value of a_{5} :

a_{2} =5a_{2-1}=5a_{1}=5 \times4=20

a_{3} =5a_{3-1}=5a_{2}=5 \times20=100

a_{4} =5a_{4-1}=5a_{3}=5 \times100=500

a_{5} =5a_{5-1}=5a_{4}=5 \times500=2500

Therefore, the value of a_{5} is 2500.

To learn more about arithmetic sequence visit:

brainly.com/question/15412619.

#SPJ1

5 0
2 years ago
Need help on 15 and 16 PLEASEE!!!
melisa1 [442]

Answer:

Part 15) The next three terms are 5/8,3/4 and 7/8

Part 16) The 37th term is -35.5

Step-by-step explanation:

we know that

In an <u>Arithmetic Sequence</u> the difference between one term and the next is a constant and this constant is called the common difference (d)

Part 15) Find the next three terms of the arithmetic sequence

\frac{1}{8},\frac{1}{4},\frac{3}{8},\frac{1}{2},...

so

a1=1/8\\ a2=1/4\\ a3=3/8\\ a4=1/2\\ a2-a1=1/4-1/8=1/8\\ a3-a2=3/8-1/4=1/8\\ a4=a3=1/2-3/8=1/8

The common difference is equal to

d=1/8

<u><em>Find the next three terms of the arithmetic sequence</em></u>

<u><em>Find a5</em></u>

a5=a4+d ----->a5=1/2+1/8=5/8

<u><em>Find a6</em></u>

a6=a5+d ----->a6=5/8+1/8=3/4

<u><em>Find a7</em></u>

a7=a6+d ----->a7=3/4+1/8=7/8

therefore

The next three terms are

5/8,3/4 and 7/8

Part 16) what is the 37th term of the arithmetic sequence

4.1,3,1.9,0.8,...?

so

a1=4.1\\ a2=3\\ a3=1.9\\ a4=0.8

a2-a1=3-4.1=-1.1\\ a3-a2=1.9-3=-1.1\\ a4=a3=0.8-1.9=-1.1

The common difference is equal to

d=-1.1

We can write an Arithmetic Sequence as a rule

an=a1+d(n-1)

substitute the values

an=4.1-1.1(n-1)

For n=37

substitute

a37=4.1-1.1(37-1)

a37=4.1-1.1(36)

a37=-35.5

4 0
3 years ago
you borrow $5,860 from a bank you will have monthly payments of $205.10 for 3 years what percent will you pay back in interest o
disa [49]

Answer:

  26%

Step-by-step explanation:

Your 36 monthly payments will total ...

  36 × $205.10 = $7383.60

$5860 of that is the amount you borrowed. The remainder is the interest you pay:

  $7383.60 -5860 = $1,523.60

As a percentage of the original loan amount this is ...

  $1523.60/$5860 × 100% = 26%

You pay back 26% of the original loan amount in interest.

3 0
2 years ago
a silkworm winds its cocoon out of one long silk fiber. to make silk thread, 3 to 10 of these silk fibers are unwound from their
Scorpion4ik [409]

Answer:

Possible lengths of the silk fibre 'x' are 100, 75, 60, 50 and 30 yards respectively.

Step-by-step explanation:

Let the length of the silk fiber = x yards.

Let the number of silk fibres = a.

It is given that the silk fibers between 3 to 10 yards are combined to form a silk thread of 300 yards.

So, we get the equation,

ax=300, where 3\leq a\leq 10

i.e. x=\frac{300}{a}, 3\leq a\leq 10

So, the possible values of 'a' are 3, 4, 5, 6 an 10.

Thus, the possible lengths of the silk fibre 'x' are 100, 75, 60, 50 and 30 yards respectively.

3 0
3 years ago
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