Comprehensive Study Guide to Arithmetic Sequences

Introduction and Definition of Number Sequences

A number sequence is defined as a collection of numbers ordered as the first, second, third, and so on, according to a specific rule. This concept often arises when computing various measurements. For example, if polygons are ordered by the number of sides—starting with triangles, then quadrilaterals, pentagons, and so forth—the sequence of the sum of their inner angles results in $180$, $360$, $540$, and so on.

Various common number patterns can be classified as sequences, including:

• Natural numbers: $1, 2, 3, \dots$
• Even numbers: $2, 4, 6, \dots$
• Odd numbers: $1, 3, 5, \dots$
• Halves of natural numbers: $1/2, 1, 1\frac{1}{2}, \dots$
• Squares of natural numbers: $1, 4, 9, \dots$

The term "sequence" originates from the Latin word sequent, meaning "following." In mathematics, it denotes objects arranged in definite positions. These objects are not limited to numbers; for instance, regular polygons, polynomials such as $1+x, 1+x^2, 1+x^3, \dots$, or even words in alphabetical order are considered sequences.

Geometric Visualizations and Computational Sequences

Geometric patterns often lead to numerical sequences. One example involves stacking dots to form triangles. By counting the number of dots in each progressive triangle, one can determine the rule required to calculate subsequent terms. Another example is the length of the hypotenuse of nested right triangles. If each new triangle is built upon the hypotenuse of the previous one with a base of $1\,\text{metre}$, the sequence of hypotenuse lengths, in metres, follows the square roots of natural numbers starting from $2$: $\sqrt{2}, \sqrt{3}, 2, \sqrt{5}, \dots$

Sequences can also be defined by algebraic properties. The sequence of natural numbers ending in $1$ can be described as $1, 11, 21, 31, \dots$. This same sequence can also be defined as the set of natural numbers that leave a remainder of $1$ when divided by $10$.

Computational tools like GeoGebra allow for the generation of sequences. For instance, to draw a sequence of regular polygons with a specified side $AB$, one can use the command Sequence[Polygon[A, B, n], n, 3, 10], where $n$ represents the number of sides ranging from $3$ to $10$. To generate numerical terms, such as the first ten squares, the command Sequence[n^2, n, 1, 10] is used. If an integer slider $m$ is used, the command Sequence[2^n, n, 1, m] will dynamically update the sequence of powers of $2$ as the value of $m$ changes.

The Principle of "Hasty Conclusions" and Circle Division

A classic cautionary tale in mathematics involves the division of a circle by points on its circumference. If $2, 3, 4, \text{ and } 5$ points are joined, the maximum number of parts created follow a doubling pattern: $2, 4, 8, \text{ and } 16$. This might lead to the hasty conclusion that the next term for $6$ points is $2^5 = 32$. However, if the points are equally spaced, $6$ points divide the circle into only $30$ parts; otherwise, the maximum is $31$. The actual formula for the maximum number of parts given $n$ points is:

$\frac{1}{24}n(n-1)(n-2)(n-3) + \frac{1}{2}n(n-1) + 1$

While this formula and the expression $2^{n-1}$ yield the same results for $n = 2, 3, 4, \text{ and } 5$, they differ significantly from $n = 6$ onwards. This highlights that the rule of formulation or the context of a sequence must be explicitly stated to determine subsequent terms with certainty.

Defining Arithmetic Sequences and Common Differences

An arithmetic sequence is a sequence that starts with a specific number and proceeds by adding a constant number again and again. In other words, in an arithmetic sequence, the same number is added to go from any term to the next, and the same number is subtracted to go from any term to the previous one.

Key terms used in this context include:

• Terms: The individual numbers that make up the sequence (e.g., first term, second term, etc.).
• Common Difference: The fixed number that is added to each term to get to the next. It is calculated by subtracting the previous term from any term (i.e., $T_n - T_{n-1} = d$).

Examples of arithmetic sequences include:

1. Even numbers ($2, 4, 6, \dots$): Starts at $2$, common difference is $2$.
2. Odd numbers ($1, 3, 5, \dots$): Starts at $1$, common difference is $2$.
3. Halves of natural numbers ($1/2, 1, 1\frac{1}{2}, \dots$): Starts at $1/2$, common difference is $1/2$.
4. Sum of outer angles of polygons ($360, 360, 360, \dots$): Starts at $360$, common difference is $0$.
5. Water tank drainage ($95, 90, 85, \dots$): Starts at $95$, common difference is $-5$. (Subtracting $5$ is equivalent to adding $-5$).

Testing and Identifying Arithmetic Sequences

To determine if a sequence is arithmetic, one must check if the difference between consecutive terms is constant. Consider the sequence of natural numbers leaving a remainder of $1$ on division by $3$: $1, 4, 7, \dots$. These can be written as: $(3 \times 0) + 1, (3 \times 1) + 1, (3 \times 2) + 1, \dots$ Subtraction of any two consecutive terms results in $(3 \times (n+1) + 1) - (3 \times n + 1) = 3$. Since the difference is constant, it is an arithmetic sequence with common difference $3$.

Conversely, the sequence of natural numbers that do NOT leave a remainder of $1$ when divided by $3$ is $2, 3, 5, 6, \dots$. Checking the differences: $3 - 2 = 1$$5 - 3 = 2$ Since the differences ($1$ and $2$) are not the same, this is not an arithmetic sequence.

Proportionality of Term and Position Changes

There is a direct relationship between the positions of terms and their values in an arithmetic sequence. In any arithmetic sequence, the change in terms is the product of the change in position and a fixed number (the common difference). This can be formally stated as: "In any arithmetic sequence, the change in terms is proportional to the change in position."

If the 3rd term of a sequence is $12$ and the 7th term is $32$:

• Position change: $7 - 3 = 4$
• Term change: $32 - 12 = 20$
• Common difference ($d$): $20 \div 4 = 5$

To find the 15th term from the 7th term:

• Position change: $15 - 7 = 8$
• Term change: $8 \times 5 = 40$
• 15th term: $32 + 40 = 72$

This proportionality can also be used to check if a specific number is a term in a sequence. For the sequence $19, 28, 37, \dots$ ($d = 9$), is $1000$ a term? We check the difference between $1000$ and the first term $19$: $1000 - 19 = 981$ Since $981 \div 9 = 109$ (an integer), $1000$ is indeed a term. Its position is $1 + 109 = 110$.

Term Connections and Sum Properties

Arithmetic sequences possess several geometric and algebraic symmetries regarding the sums of their terms:

1. Consecutive Terms Symmetry: In an arithmetic sequence, the sum of terms just before and just after a specific term is twice that term ($T_{n-1} + T_{n+1} = 2T_n$). Consequently, the sum of three consecutive terms is three times the middle term.

2. Odd Number of Consecutive Terms: The sum of an odd number of consecutive terms is the product of the middle term and the count of terms. For example, if the 10th term is $25$, the sum of the 9th, 10th, and 11th terms is $3 \times 25 = 75$. The sum of five terms from 8th to 12th is $5 \times 25 = 125$.

3. Equidistant Pairs: The sum of two terms at the same distance behind and ahead of a central term is twice that central term. In more general terms, if the sum of two positions is equal to the sum of two other positions ($i + j = k + l$), then the sum of the terms at those positions is the same ($T_i + T_j = T_k + T_l$).

For example, if the sum of the 2nd and 5th terms is $35$ ($2 + 5 = 7$):

• The sum of the 3rd and 4th terms ($3 + 4 = 7$) must also be $35$.
• The sum of the 1st and 6th terms ($1 + 6 = 7$) must also be $35$.

Practical Problems and Real-World Applications

These principles can solve complex problems regarding geometric shapes and series:

• Pentagon Angles: The sum of inner angles of a pentagon is $540^\circ$. If the angles are in an arithmetic sequence, the middle (3rd) angle is $540 \div 5 = 108^\circ$. If the smallest angle is $40^\circ$, one can determine the common difference and the other angles.
• Series Sums: If the sum of the first five terms of a sequence is $250$, the 3rd (middle) term is $250 \div 5 = 50$. If the first term is $10$, the common difference is $(50 - 10) \div 2 = 20$, resulting in the sequence $10, 30, 50, 70, 90$.
• Reverse Calculations: Knowing the sum of terms like the 11th and 21st ($11 + 21 = 32$) allows one to find the 16th term, as $16 + 16 = 32$. Since the sum of positions is the same, $T_{11} + T_{21} = T_{16} + T_{16} = 2T_{16}$. If the sum is $80$, the 16th term is $40$.