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study each term of the sequence as it compares to its term number. then the following questions: a. is it multiple of the term number? b. is it a multiple of the term number, there will be a common number.
Question about the sequence
- Thread starter Haleluya
- Start date
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To answer your questions about the sequence, let's take a closer look at each term in relation to its term number.
Let's assume the sequence is defined as follows: a₁, a₂, a₃, a₄, ...
a. Is each term a multiple of its term number?
To determine this, we need to check if a term aₙ is divisible by its corresponding term number, n.
If aₙ is a multiple of n, it means that aₙ ÷ n gives an integer value with no remainder. In other words, n evenly divides aₙ.
For example, let's consider the sequence: 2, 8, 15, 24, 35, ...
The first term, 2, is divisible by 1 since 2 ÷ 1 = 2 with no remainder.
The second term, 8, is divisible by 2 since 8 ÷ 2 = 4 with no remainder.
The third term, 15, is not divisible by 3 since 15 ÷ 3 = 5 with no remainder.
The fourth term, 24, is divisible by 4 since 24 ÷ 4 = 6 with no remainder.
The fifth term, 35, is not divisible by 5 since 35 ÷ 5 = 7 with no remainder.
So, in this example, not every term is a multiple of its term number.
b. Is there a common number if all terms are multiples of their term number?
If all terms of the sequence are multiples of their term number, it implies that there is a common factor shared by all the terms.
For example, let's consider the sequence: 3, 6, 9, 12, 15, ...
In this sequence, each term is divisible by its term number (e.g., 3 is divisible by 1, 6 is divisible by 2, 9 is divisible by 3, etc.), so there is indeed a common number, which is the term number itself. In this case, the common number is the term number itself.
However, it's important to note that not all sequences will have a common number if all terms are multiples of their term number. It depends on the specific pattern or rule governing the sequence.
Let's assume the sequence is defined as follows: a₁, a₂, a₃, a₄, ...
a. Is each term a multiple of its term number?
To determine this, we need to check if a term aₙ is divisible by its corresponding term number, n.
If aₙ is a multiple of n, it means that aₙ ÷ n gives an integer value with no remainder. In other words, n evenly divides aₙ.
For example, let's consider the sequence: 2, 8, 15, 24, 35, ...
The first term, 2, is divisible by 1 since 2 ÷ 1 = 2 with no remainder.
The second term, 8, is divisible by 2 since 8 ÷ 2 = 4 with no remainder.
The third term, 15, is not divisible by 3 since 15 ÷ 3 = 5 with no remainder.
The fourth term, 24, is divisible by 4 since 24 ÷ 4 = 6 with no remainder.
The fifth term, 35, is not divisible by 5 since 35 ÷ 5 = 7 with no remainder.
So, in this example, not every term is a multiple of its term number.
b. Is there a common number if all terms are multiples of their term number?
If all terms of the sequence are multiples of their term number, it implies that there is a common factor shared by all the terms.
For example, let's consider the sequence: 3, 6, 9, 12, 15, ...
In this sequence, each term is divisible by its term number (e.g., 3 is divisible by 1, 6 is divisible by 2, 9 is divisible by 3, etc.), so there is indeed a common number, which is the term number itself. In this case, the common number is the term number itself.
However, it's important to note that not all sequences will have a common number if all terms are multiples of their term number. It depends on the specific pattern or rule governing the sequence.
OP
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- Thread Starter
- #3
examine the sequence. does it increase or decrease?
a. if it increases slowly, consider expressions that involve the term number plus or minus a constant like: n + 2 or n - 3.
b. if it increases moderately, think about multiples of the term number plus or minus a constant like: 2n or 3n - 1.
c. if the sequence increases very rapidly, try powers of the term number plus or minus a constant like n^2 or (n^2) + 1.
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To determine how the sequence increases or decreases, we can analyze the pattern based on the given suggestions:
a. If the sequence increases slowly:
In this case, the terms would be closely related to the term number, with a small constant added or subtracted from it. Examples of such expressions include:
- n + 2
- n - 3
These expressions indicate that the sequence is increasing gradually with each term.
b. If the sequence increases moderately:
If the sequence exhibits a moderate increase, the terms may be multiples of the term number with a constant added or subtracted. Examples include:
- 2n
- 3n - 1
These expressions suggest that the sequence is increasing at a slightly faster rate than in the previous case.
c. If the sequence increases very rapidly:
If the sequence shows a rapid increase, it may involve powers of the term number plus or minus a constant. Examples include:
- n^2
- (n^2) + 1
Expressions like these indicate that the sequence is growing at an accelerated rate, with each term substantially larger than the previous one.
To accurately determine how a specific sequence increases or decreases, it would be helpful to provide the actual sequence or any additional information about the terms involved.
OP
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- Thread Starter
- #5
if the sequence consists of fraction, examine how the denominator and numerator changes as separate sequences. for example: an = (n + 1)/(n^2) yieleds.
the sequence: 2/1, 3/4, 4/9, 5/16, 6/25, ...
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To analyze the given sequence of fractions, which follows the pattern an = (n + 1)/(n^2), let's examine how the numerator and denominator change as separate sequences.
Numerator sequence:
The numerator represents the "n + 1" part of the fraction. For each term in the sequence, the numerator increases by 1.
Numerator sequence: 2, 3, 4, 5, 6, ...
Denominator sequence:
The denominator represents the "n^2" part of the fraction. For each term in the sequence, the denominator is equal to the square of the term number (n^2).
Denominator sequence: 1, 4, 9, 16, 25, ...
Now, let's observe the values of each sequence when combined:
Term 1:
Numerator: 2
Denominator: 1
Fraction: 2/1
Term 2:
Numerator: 3
Denominator: 4
Fraction: 3/4
Term 3:
Numerator: 4
Denominator: 9
Fraction: 4/9
Term 4:
Numerator: 5
Denominator: 16
Fraction: 5/16
Term 5:
Numerator: 6
Denominator: 25
Fraction: 6/25
From this analysis, we can observe that as the term number increases, the numerator increases by 1, while the denominator follows the sequence of perfect squares.
Regarding the overall pattern of the sequence, the fractions are becoming smaller as the term number increases. Additionally, the denominator sequence increases faster than the numerator sequence, resulting in decreasing values for the fractions.
It's important to note that this sequence is specific to the given formula, and different formulas may lead to different patterns and behaviors.