Sequence mathematics help

In summary, the powers of numbers in the given sequence that will always be in the sequence are the odd numbers. This can be proven by setting the members of the sequence to a power of a number in the sequence and observing that the last term is always divisible by 2. Additionally, by using modular arithmetic, it can be shown that the sequence is always equivalent to -1 mod 3, and therefore, the even powers do not lie in the sequence.
  • #1
Natasha1
493
9
Could anyone just run through my answer to this question and spote any mistakes, to let me know if I have done this correctly please? Thanks!

Question:

Which powers of numbers in the sequence are always in the sequence and which are not. Prove your findings.

sequence 2, 5, 8, 11, 14...


Answer:

a_n = 3n+2

So a member in the sequence taken to a power of a number in the sequence will be of the form (3n+2)^x where x is any member of the sequence.

So

(3n+2)^x = (3n)^x + (3n)^x-1 *2^1 + (3n)^x-2 * 2^2 +...+(3n)^1 * 2^x-1 + 2^x

Noticing from this that the last term is always 2^x.

If we set (3x+2)^x = 3q+2

Solving for q:

q= (2^x - 2)/3

As q has to be an integer, x has to take values for which (2^x - 2)/3 remains an integer are the odd numbers (1, 3, 5, 7, ...)

Therefore, the powers of the numbers which will always be in the sequence are the odd numbers. All the even powers do not lie in the sequence.
 
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  • #2
Natasha1 said:
Could anyone just run through my answer to this question and spote any mistakes, to let me know if I have done this correctly please? Thanks!

Question:

Which powers of numbers in the sequence are always in the sequence and which are not. Prove your findings.

sequence 2, 5, 8, 11, 14...


Answer:

a_n = 3n+2

So a member in the sequence taken to a power of a number in the sequence will be of the form (3n+2)^x where x is any member of the sequence.

So

(3n+2)^x = (3n)^x + (3n)^x-1 *2^1 + (3n)^x-2 * 2^2 +...+(3n)^1 * 2^x-1 + 2^x

Noticing from this that the last term is always 2^x.

If we set (3x+2)^x = 3q+2
You've expanded the terms out incorrectly. Where are all the binomial coefficients?
[tex](x + y) ^ n = \sum_{k = 0} ^ n \left( \begin{array}{l} n \\ k \end{array} \right) x ^ k y ^ {n - k}[/tex]
Solving for q:

q= (2^x - 2)/3
Nope, this is wrong.
[tex]q = \frac{(3x + 2) ^ x - 2}{3}[/tex]
Expand (3x + 2)x out like above, we notice that all but the last term is divisible by 3, we have:
[tex]q = \frac{(3x + 2) ^ x - 2}{3} = A + \frac{2 ^ n - 2}{3}[/tex], where A is some positive integer.
As q has to be an integer, x has to take values for which (2^x - 2)/3 remains an integer are the odd numbers (1, 3, 5, 7, ...)

Therefore, the powers of the numbers which will always be in the sequence are the odd numbers. All the even powers do not lie in the sequence.
Why? Can you prove that: 22n + 1 - 2, where n is an integer is divisible by 3?
---------------------
Hint, another way is that:
[tex]a_n \equiv -1 \mbox{ mod } 3[/tex]
[tex]\Rightarrow (a_n) ^ {2k} \equiv ? \mbox{ mod } 3, \ k \in \mathbb{N}[/tex]
[tex]\Rightarrow (a_n) ^ {2k + 1} \equiv ? \mbox{ mod } 3, \ k \in \mathbb{N}[/tex]
Can you go from here? :)
 
Last edited:

1. What is sequence mathematics?

Sequence mathematics is a branch of mathematics that deals with the study of sequences, which are ordered lists of numbers or other mathematical objects. These sequences can follow a specific pattern or rule, and can be used to solve a variety of mathematical problems.

2. What are the different types of sequences?

There are several types of sequences, including arithmetic sequences, geometric sequences, and recursive sequences. Arithmetic sequences follow a specific pattern where each term is obtained by adding a constant value to the previous term. Geometric sequences follow a pattern where each term is obtained by multiplying the previous term by a constant value. Recursive sequences use a rule or equation to generate each term based on the previous terms in the sequence.

3. How are sequences used in mathematics?

Sequences are used in a variety of mathematical fields, including number theory, calculus, and statistics. They can be used to model real-life situations, solve equations, and prove mathematical theorems. In addition, sequence patterns and relationships can be used to make predictions and make sense of numerical data.

4. What is the difference between a sequence and a series?

A sequence is an ordered list of numbers or objects, while a series is the sum of the terms in a sequence. In other words, a series is the result of adding all the terms in a sequence together. For example, the sequence 1, 2, 3, 4, 5 has a series of 15 (1+2+3+4+5).

5. How do I find the next term in a sequence?

To find the next term in a sequence, you can look for a pattern or rule that each term follows. This could involve adding a constant value, multiplying by a constant value, or using a recursive formula. Once you have identified the pattern, you can use it to predict the next term in the sequence.

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