What is the Mystery Number Sequence?

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In summary, the multiplication of the 1st and 2nd digits gives an answer of 4 and the multiplication of the 3rd and 4th digits gives 70. To solve for the unknown digits, a quadratic formula can be used to express d in terms of A1. Substituting this back into the other equation and simplifying results in a fourth-order polynomial, which can be solved numerically. Alternatively, using a geometric series instead of an arithmetic series makes the problem much easier to solve.
  • #1
devanlevin
in a sequence, it is known that the multiplication of the 1st and second digits gives an answer or 4, the multiplication of the 3rd and 4th gives 70

A1*A2=A1*(A1+d)=4
A3*A4=(A1+2d)(A1+3d)=70

A1^2+A1d=4

A1^2+3A1d+2A1d+6d^2=70
A1^2+5A1d+6d^2=70

A1^2=4-A1d

4-A1d+5A1d+6d^2=70
4A1d+6d^2=66
3d^2+2A1d=33

from here i don't know what to do,
 
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  • #2
"from here i don't know what to do"

Solve it as a quadratic to get a solution in the form of d as a function of A1. BTW this means there is not just one solution but a whole family of them.

BTW. Is there any particular reason that you chose to use an arithmetic series? It's dead easy if you choose to use a geometric series.
 
  • #3
how do i do that?
 
  • #4
uart said:
"from here i don't know what to do"

Solve it as a quadratic to get a solution in the form of d as a function of A1. BTW this means there is not just one solution but a whole family of them.

Incorrect.

You have two equations in two unkwowns; you may reduce the system to one unknown being the solution of a fourth-order polynomial, yielding 4 solutions to the problem.
 
  • #5
devanlevin said:
how do i do that?

Do what, solve the quadratic or use a geometric series instead of an arithmetic series?

To finish your current solution solve your last equation (3d^2+2A1d-33=0) using the quadratic formula to express d in terms of A1. Then continue (as arildno corrected me) and subtitute that back into the other equation (A1^2=4-A1d) to get an equation in terms of A1 alone. Solve this equation by re-arranging to get the sqare-root term on one side and all other terms on the other side of the equals. Now square both sides to get a 4th order polynomal which may require numerical methods to solve.

Alternatively if there is no particular reason for using an arithmetic series instead of a gemetric series then just use the sequence a, ar, ar^2, ar^3 etc which makes the problem trivial to solve. (you get r^4 = 70/4 with very little effort).

BTW. You still haven't told me why you feel that you should use an arithmetic series?
 
Last edited:
  • #6
As it happens, you'll get "lucky", in that the resulting fourth-order polynomial is really a second-order polynomial in the variable A^2. That can be readily solved, and if you are really good, your solutions will agree with mine:
[tex]A_{1}^{(1)}=1,d^{(1)}=3[/tex]
[tex]A_{1}^{(2)}=-1,d^{(2)}=-3[/tex]
[tex]A_{1}^{(3)}=4\sqrt{3},d^{(3)}=-\frac{11}{\sqrt{3}}[/tex]
[tex]A_{1}^{(4)}=-4\sqrt{3},d^{(4)}=\frac{11}{\sqrt{3}}[/tex]
 

What is a number sequence?

A number sequence is a list of numbers that follow a specific pattern or rule. The numbers can either increase or decrease in a specific pattern, or they can be a random arrangement of numbers.

What are the different types of number sequences?

There are several types of number sequences, including arithmetic sequences where the difference between each number is constant, geometric sequences where the ratio between each number is constant, and Fibonacci sequences where each number is the sum of the two previous numbers.

How do you find the rule of a number sequence?

To find the rule of a number sequence, you need to identify the pattern or relationship between each number. This can involve looking at the difference between each number, the ratio between each number, or any other pattern that the numbers follow. Once you have identified the pattern, you can use it to predict the next numbers in the sequence.

Why are number sequences important?

Number sequences are important because they help us understand and predict patterns in mathematics and in the world around us. They are also used in various fields of science, such as in genetics and physics, to model and understand complex systems.

How can number sequences be used in everyday life?

Number sequences can be used in various ways in everyday life, such as in budgeting and financial planning, analyzing data in business and economics, and even in sports and games. They can also be used to solve problems and make predictions in various fields, such as weather forecasting and stock market analysis.

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