In: Advanced Math

Given the series of numbers:

{1.,0.612547,0.466856,0.375214,0.311459,0.264691,0.229088,0.201197,0.17884,0.160573,0.145406,0.132639,0.121764,0.112404,0.104273,0.0971538,0.0908741,0.0852989,0.08032,0.0758497,0.0718166,0.0681615,0.0648355,0.0617975,0.0590128,0.056452,0.05409,0.0519052,0.0498791,0.0479955,0.0462403,0.0446012,0.0430675,0.0416295,0.0402789,0.0390081,0.0378105,0.0366801,0.0356115,0.0346001,0.0336414,0.0327315}

What is the 100th term to 12-decimal places? How is this sequence made?

Solution:

%plot of the given data and fitted curve in matlab

close all

clear all

y=[1,0.612547,0.466856,0.375214,0.311459,0.264691,0.229088,...

0.201197,0.17884,0.160573,0.145406,0.132639,0.121764,0.112404,...

0.104273,0.0971538,0.0908741,0.0852989,0.08032,0.0758497,0.0718166,0.0681615,0.0648355,0.0617975,0.0590128,0.056452,0.05409,...

0.0519052,0.0498791,0.0479955,0.0462403,0.0446012,0.0430675,...

0.0416295,0.0402789,0.0390081,0.0378105,0.0366801,0.0356115,...

0.0346001,0.0336414,0.0327315];

x=1:length(y);

p = polyfit(x,y,5) %polynomial of degree 5 fit of the data

y1 = polyval(p,x);

figure

plot(x,y,'o')

hold on

plot(x,y1)

hold off

%output

p = Columns 1 through 5: -0.00000019943 0.00002447160 -0.00113191903 0.02452586379 -0.25205830228 Column 6: 1.10494242333

hence the general term of the given sequence

substituing n =100 to get 100th term of the sequence is given by

-457.9455002640371

Assignment: Enter a linear series of numbers 1, 2, 3, . . . . .
100 in column A from A1 to A100. Enter a second series of numbers
20, 22, 24, . . . . . 218 in Column “B” from B1 to B100. Plot
Column “B” as a function of column “A”. Use a linear trend-line and
find the equation of the line. Specify the slope and the
y-intercept of the line. Save your file and submit it...

The Fibonacci sequence is the series of numbers 0, 1, 1,
2, 3, 5, 8,.... Formally, it can be expressed as:
fib0 = 0
fib1 = 1
fibn = fibn-1 + fibn-2
Write a multithreaded C++ program that generates the
Fibonacci series using the pthread library. This program should
work as follows: The user will enter on the command line the number
of Fibonacci numbers that the program will generate. The program
will then create a separate thread that will...

Python: Using Jupyter Notebook
1. Write code to generate Fibonacci series.
Fibonacci numbers – 1, 1, 2, 3, 5, 8, …
2. Check if a number is an Armstrong number
A positive integer is called an Armstrong number of order n
if
abcd... = a^n + b^n + c^n + d^n + ...
In case of an Armstrong number of 3 digits, the sum of cubes of
each digits is equal to the number itself. For example:
153 = 1*1*1...

1. Given the series:
∞∑k=1 2/k(k+2)
does this series converge or diverge?
converges
diverges
If the series converges, find the sum of the series:
∞∑k=1 2/k(k+2)=
2. Given the series:
1+1/4+1/16+1/64+⋯
does this series converge or diverge?
diverges
converges
If the series converges, find the sum of the series:
1+1/4+1/16+1/64+⋯=

1.write a small program using a loop to add a series of
numbers
2.write a function called "main" that performs several given
steps. Be sure to call the main() function so that its code
executes
In python and doesn't have to be long. just long enough to do
what it says. Thank you.

Using Python:
1. Compute the difference of differences between consecutive
numbers of a series:
input
ser = pd.Series([1, 3, 6, 10, 15, 21, 27, 35])
output:
[nan, 2.0, 3.0, 4.0, 5.0, 6.0, 6.0, 8.0]
[nan, nan, 1.0, 1.0, 1.0, 1.0, 0.0, 2.0]
2. Compute the euclidean distance between two series:
Input:
p = pd.Series([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
q = pd.Series([10, 9, 8, 7, 6, 5, 4, 3, 2, 1])
Desired Output:
18.165

1. Prove that given n + 1 natural numbers, there are always two
of them such that their difference is a multiple of n.
2. Prove that there is a natural number composed with the digits
0 and 5 and divisible by 2018.
both questions can be solved using pigeonhole principle.

The Fibonacci Sequence is a series of integers. The first two
numbers in the sequence are both 1; after that, each number is the
sum of the preceding two numbers.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
For example, 1+1=2, 1+2=3, 2+3=5, 3+5=8, etc.
The nth Fibonacci number is the nth number in this sequence, so
for example fibonacci(1)=1, fibonacci(2)=1, fibonacci(3)=2,
fibonacci(4)=3, etc. Do not use zero-based counting; fibonacci(4)is
3, not 5.
Your assignment...

Given the following series: 1, 2, 5, 26, 677, ….. such that the
nth term of the series equals to (n-1)th ^2 +1 and the first term
of the series is 1. Write a C program using recursive function
named f to compute the nth term. Use for loop to print the values
of first n terms in the series. You will take input n from the
user.
Example output:
Enter number of terms: *value*
*numbers here*
nth term: *value*...

Given the following series,
tn =
10tn-1+1
where tn is the current
term and tn-1 is the previous terms.
Write a Matlab function, which will
receive t n-1 will calculate and return
tn.
Use a for loop to call the function in (a) and print the first
10 terms computed by tn for n=1,…,10. Assume that
t0 = 0.
Write a Matlab script that loops until the sum of series is
12345 and display the numbers of terms added to...

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