1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Ainat [17]
3 years ago
15

2,17,82,257,626,1297 next one please ?​

Mathematics
1 answer:
In-s [12.5K]3 years ago
4 0

The easy thing to do is notice that 1^4 = 1, 2^4 = 16, 3^4 = 81, and so on, so the sequence follows the rule n^4+1. The next number would then be fourth power of 7 plus 1, or 2402.

And the harder way: Denote the <em>n</em>-th term in this sequence by a_n, and denote the given sequence by \{a_n\}_{n\ge1}.

Let b_n denote the <em>n</em>-th term in the sequence of forward differences of \{a_n\}, defined by

b_n=a_{n+1}-a_n

for <em>n</em> ≥ 1. That is, \{b_n\} is the sequence with

b_1=a_2-a_1=17-2=15

b_2=a_3-a_2=82-17=65

b_3=a_4-a_3=175

b_4=a_5-a_4=369

b_5=a_6-a_5=671

and so on.

Next, let c_n denote the <em>n</em>-th term of the differences of \{b_n\}, i.e. for <em>n</em> ≥ 1,

c_n=b_{n+1}-b_n

so that

c_1=b_2-b_1=65-15=50

c_2=110

c_3=194

c_4=302

etc.

Again: let d_n denote the <em>n</em>-th difference of \{c_n\}:

d_n=c_{n+1}-c_n

d_1=c_2-c_1=60

d_2=84

d_3=108

etc.

One more time: let e_n denote the <em>n</em>-th difference of \{d_n\}:

e_n=d_{n+1}-d_n

e_1=d_2-d_1=24

e_2=24

etc.

The fact that these last differences are constant is a good sign that e_n=24 for all <em>n</em> ≥ 1. Assuming this, we would see that \{d_n\} is an arithmetic sequence given recursively by

\begin{cases}d_1=60\\d_{n+1}=d_n+24&\text{for }n>1\end{cases}

and we can easily find the explicit rule:

d_2=d_1+24

d_3=d_2+24=d_1+24\cdot2

d_4=d_3+24=d_1+24\cdot3

and so on, up to

d_n=d_1+24(n-1)

d_n=24n+36

Use the same strategy to find a closed form for \{c_n\}, then for \{b_n\}, and finally \{a_n\}.

\begin{cases}c_1=50\\c_{n+1}=c_n+24n+36&\text{for }n>1\end{cases}

c_2=c_1+24\cdot1+36

c_3=c_2+24\cdot2+36=c_1+24(1+2)+36\cdot2

c_4=c_3+24\cdot3+36=c_1+24(1+2+3)+36\cdot3

and so on, up to

c_n=c_1+24(1+2+3+\cdots+(n-1))+36(n-1)

Recall the formula for the sum of consecutive integers:

1+2+3+\cdots+n=\displaystyle\sum_{k=1}^nk=\frac{n(n+1)}2

\implies c_n=c_1+\dfrac{24(n-1)n}2+36(n-1)

\implies c_n=12n^2+24n+14

\begin{cases}b_1=15\\b_{n+1}=b_n+12n^2+24n+14&\text{for }n>1\end{cases}

b_2=b_1+12\cdot1^2+24\cdot1+14

b_3=b_2+12\cdot2^2+24\cdot2+14=b_1+12(1^2+2^2)+24(1+2)+14\cdot2

b_4=b_3+12\cdot3^2+24\cdot3+14=b_1+12(1^2+2^2+3^2)+24(1+2+3)+14\cdot3

and so on, up to

b_n=b_1+12(1^2+2^2+3^2+\cdots+(n-1)^2)+24(1+2+3+\cdots+(n-1))+14(n-1)

Recall the formula for the sum of squares of consecutive integers:

1^2+2^2+3^2+\cdots+n^2=\displaystyle\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}6

\implies b_n=15+\dfrac{12(n-1)n(2(n-1)+1)}6+\dfrac{24(n-1)n}2+14(n-1)

\implies b_n=4n^3+6n^2+4n+1

\begin{cases}a_1=2\\a_{n+1}=a_n+4n^3+6n^2+4n+1&\text{for }n>1\end{cases}

a_2=a_1+4\cdot1^3+6\cdot1^2+4\cdot1+1

a_3=a_2+4(1^3+2^3)+6(1^2+2^2)+4(1+2)+1\cdot2

a_4=a_3+4(1^3+2^3+3^3)+6(1^2+2^2+3^2)+4(1+2+3)+1\cdot3

\implies a_n=a_1+4\displaystyle\sum_{k=1}^3k^3+6\sum_{k=1}^3k^2+4\sum_{k=1}^3k+\sum_{k=1}^{n-1}1

\displaystyle\sum_{k=1}^nk^3=\frac{n^2(n+1)^2}4

\implies a_n=2+\dfrac{4(n-1)^2n^2}4+\dfrac{6(n-1)n(2n)}6+\dfrac{4(n-1)n}2+(n-1)

\implies a_n=n^4+1

You might be interested in
What is number in unit form ​
Mazyrski [523]
What are the answer choices can you attach a pic?
5 0
3 years ago
Taryn's dog, Skeeter, consumes an average of 1,155 calories per day. However, Skeeter participated in a dog race this morning, s
trapecia [35]

Answer:

The answer is C. x = 2

3 0
3 years ago
The average systolic blood pressure was found to be 140 mm for people who have high responsibility jobs, specifically investment
defon

Answer:

t = -1.428

Step-by-step explanation:

Average blood pressure among people with high responsibility jobs = u = 140 mm

Number of people sampled for the study = n = 51

Average blood pressure of the sample = x = 139 mm

Standard deviation of the sample = s = 5 mm

The claim is that practicing yoga decreases the blood pressure. So we have to test that after practicing yoga, the average blood pressure will be lesser than 140. Based on this we can formulate the null and alternate hypothesis as:

Null Hypothesis: \mu \geq 140

Alternate Hypothesis: \mu < 140

Since population standard deviation is not known, we cannot use z-distribution for this test. We are given the value of sample standard deviation, so t-test will be used for this case.

Test statistic is calculated as:

t=\frac{x-u}{\frac{s}{\sqrt{n} } }

Using the values in this formula gives us:

t=\frac{139-140}{\frac{5}{\sqrt{51}}}\\\\ t=-1.428

Therefore, the value of test statistic is -1.428

4 0
3 years ago
Julie is digging a square area in her backyard a for a vegtable garden. If the area is 52 square feet, what is the approximate l
AURORKA [14]

Answer:

<em>The correct option will be:   (b) 7 feet.</em>

Step-by-step explanation:

Suppose, the length of the garden is  x feet.

As the <u>shape of the garden is like a square, then the area of the garden</u> will be: (length)^2 = x^2 feet²

Given that, the <u>area is 52 square feet</u>. So, the equation will be.......

x^2= 52\\ \\ x=\sqrt{52} = 7.211.... \approx 7

So, the approximate length of the garden will be 7 feet.

4 0
3 years ago
Jesse takes his dog and cat for their annual vet visit.Jesse's dog weighs 23 pounds.The vet tells him his cat's weight is 5/8 as
Kobotan [32]
If I’m not mistaken the cats weight is 14.4
5 0
3 years ago
Other questions:
  • Simplify the expression : l-8l + l8l
    13·1 answer
  • a rectangular field has an area of 1,764m^2. the width of the field is 13m more then the length. what is the perimeter of the fi
    12·1 answer
  • Always sometimes never: two angles of a triangle add up to to 90°
    8·1 answer
  • What are the next two terms in the pattern 3,6,5,10,9,18,17
    13·1 answer
  • If a is 2 inches long, b is 5 inches long, and c is 12 inches long, what is the total volume of the two blocks?
    11·1 answer
  • 6. 1 point Mark only one oval. Not biased Bias 7. 2 points Mark only one oval. 26 oz 27 oz 28 oz A town has 15,000 registered vo
    5·1 answer
  • Which inequality will have a shaded area below the boundary line?
    7·1 answer
  • The picture is up there thank you:))
    13·1 answer
  • Commutative property under integers and rational numbers with examples​
    13·1 answer
  • Find the volume of the sphere d=18
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!