Notes on Arithmetic Sequences, Geometric Sequences, Series, and Polynomial Division

Sequence Basics

• A sequence is a set of values that follow a particular order; it can be finite or infinite. Each value is a term.
• Finite sequence: has a first term and a last term. Example: 3, 6, 9, 12, 15. First term = 3, last term = 15.
• Infinite sequence: has no last term. Example: 2, 4, 8, 16, 32, … (last term cannot be determined).
• nth term (an) of a sequence is the term in position n. For the sequence 1, 3, 5, 7, 9, the nth term can be written as
  $a_n = 2n - 1.$ This is the general term of the sequence.
• To find a specific term, substitute n with the desired position. For example, to find the 12th term:
  $a_{12} = 2(12) - 1 = 24 - 1 = 23.$ So the 12th term is 23.

General Term of a Sequence

• The square numbers form a pattern: 1, 4, 9, 16, 25, …
• These are given by the squares of positive integers:
  1^2, 2^2, 3^2, 4^2, 5^2,
  
  so the general term is $a_n = n^2.$
• Sequences can be described by two kinds of formulas:
  • Explicit formula: gives the nth term directly in terms of n. Examples: $a<em>n = 2n - 1,\, a</em>n = n^2.$
  • Recursive formula: defines each term in relation to preceding terms and a seed (the first term).
  • For the sequence with explicit formula $a<em>n = 2n - 1$, a recursive form can be written as a1 = 1,
    \ an = a{n-1} + 2 \, (n > 1).
• Series (sum of terms) concept:
  • A series is the sum of the terms of a sequence: $a<em>1 + a</em>2 + a<em>3 + \cdots + a</em>n.$
  • Examples (sums of first five terms):
  • For the sequence 1, 3, 5, 7, 9: $1 + 3 + 5 + 7 + 9 = 25.$
  • For the sequence 1, 4, 9, 16, 25: $1 + 4 + 9 + 16 + 25 = 55.$

Explicit Term Finding: Example Problems

• Example: Find the general term for the sequence 9, 11, 13, 15, 17, 19, …
  • The general term is linear: assume $a_n = an + b.$
  • Use two terms to form equations:
  • For n = 1: $a + b = 9.$
  • For n = 2: $2a + b = 11.$
  • Subtract equations: $(2a + b) - (a + b) = 11 - 9 \Rightarrow a = 2.$
  • Substitute to find b: $2 + b = 9 \Rightarrow b = 7.$
  • Therefore, $a_n = 2n + 7.$

Arithmetic Sequences

• An arithmetic sequence (arithmetic progression) has a constant difference d between consecutive terms.
  • If d > 0, the sequence is increasing; if d < 0, it is decreasing.
  • If the first term is $a<em>1$, the sequence is $a</em>1,\ a<em>1 + d,\ a</em>1 + 2d,\ a<em>1 + 3d,\ a</em>1 + 4d, \dots$
• General term (nth term) of an arithmetic sequence:
  $a<em>n = a</em>1 + (n - 1)d.$
• The common difference can be found from two terms:
  $d = \frac{a<em>n - a</em>1}{n - 1}.$
• Let's Practice: Example 1
  • Find the 16th term of the sequence 2, 6, 10, 14, …
  • Here, $a_1 = 2$ and the common difference is $d = 6 - 2 = 4.$
  • Use the formula: $a<em>n = a</em>1 + (n - 1)d.$
  • Then
    $a_{16} = 2 + (16 - 1)\cdot 4 = 2 + 15\cdot 4 = 2 + 60 = 62.$
• Example 3 (Two-term system to find a1 and d):
  • Given that in an arithmetic sequence the 3rd term is 27 and the 5th term is 41, find the 8th term.
  • Write equations:
  • $a<em>3 = a</em>1 + 2d = 27.$
  • $a<em>5 = a</em>1 + 4d = 41.$
  • Solve: subtracting gives $2d = 14 \Rightarrow d = 7.$
  • Then $a_1 = 27 - 2\cdot 7 = 13.$
  • The 8th term: $a<em>8 = a</em>1 + 7d = 13 + 7\cdot 7 = 13 + 49 = 62.$

Distinguishing Arithmetic vs Geometric Sequences

• Check differences vs ratios:
  • Arithmetic: consecutive differences are constant. That is, compute $a<em>2 - a</em>1,\ a<em>3 - a</em>2,\ a<em>4 - a</em>3,\dots$; if all are equal, the sequence is arithmetic with common difference $d$.
  • Geometric: consecutive ratios are constant. That is, compute $\frac{a<em>2}{a</em>1},\ \frac{a<em>3}{a</em>2},\ \frac{a<em>4}{a</em>3},\dots$; if all are equal, the sequence is geometric with common ratio $r$.
• Examples:
  • The sequence 5, 9, 13, 17, … has differences 4, 4, 4, … → arithmetic with $d=4$ (increasing).
  • The sequence 4, 2, 1, … has ratios $a<em>2/a</em>1 = 1/2, a<em>3/a</em>2 = 1/2, \dots$ → geometric with $r = \tfrac{1}{2}$.
• Formulas:
  • Arithmetic nth term: $a<em>n = a</em>1 + (n-1)d.$
  • Geometric nth term: $a<em>n = a</em>1 r^{n-1}.$

Mixed Practice: Arithmetic vs Geometric (Summary)

• A sequence is arithmetic if it has a common difference; geometric if it has a common ratio.
• The nth term formulas are fundamental tools:
  • Arithmetic: $a<em>n = a</em>1 + (n-1)d.$
  • Geometric: $a<em>n = a</em>1 r^{n-1}.$
• Sign of d or r matters for monotonicity and sign behavior.

Geometric Series

• A geometric series is the sum of a geometric sequence:
  $S<em>n = a</em>1 + a<em>1 r + a</em>1 r^2 + \cdots + a_1 r^{n-1}.$
• Closed form (for r ≠ 1):
  $S<em>n = \frac{a</em>1(1 - r^n)}{1 - r}.$
• Example: Let $a<em>1 = 1,\ r = 2,\ n = 9.$ Then $S</em>9 = \frac{1(1 - 2^9)}{1 - 2} = \frac{1 - 512}{-1} = 511.$
• Infinite geometric series (|r| < 1):
  • If the series is infinite, it converges to
    $S<em>\infty = \frac{a</em>1}{1 - r}.$
  • Intuition: as n grows, the terms get small when |r| < 1.

Arithmetic Series

• The sum of the first n terms of an arithmetic sequence is called an arithmetic series. If the first term is $a<em>1$, the last term is $a</em>n$, and the number of terms is $n$, then
  $S<em>n = \frac{n}{2} (a</em>1 + a_n).$
• If the last term is not readily given, use
  $S<em>n = \frac{n}{2} [2a</em>1 + (n - 1)d],$
  where $d$ is the common difference.
• Example: Find the sum of the first 12 terms in the sequence 1, 2, 3, 4, …
  • This is arithmetic with $a<em>1 = 1,\ d = 1.$ Then $a</em>{12} = a_1 + (12 - 1)d = 12.$
  • Sum: $S<em>{12} = \frac{12}{2} (a</em>1 + a_{12}) = 6(1 + 12) = 78.$
  • Alternative formula: using $S<em>n = \frac{n}{2} [2a</em>1 + (n - 1)d]$ gives the same result: $S_{12} = \frac{12}{2} [2\cdot 1 + (12 - 1)\cdot 1] = 6(2 + 11) = 78.$

Synthetic Division

• Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form $x - k$.
• Basic procedure (outline):
  1) Ensure divisor is of the form $x - k$; identify $k$.
  2) Write the coefficients of the dividend in descending powers of x.
  3) Bring down the leading coefficient to the bottom row.
  4) Multiply it by $k$ and write under the next coefficient; add the column to get the next bottom entry.
  5) Repeat until all columns are processed.
  6) The bottom row (except the final entry) gives the quotient coefficients; the last entry is the remainder.
  7) The exact division result can be written as the quotient plus remainder divided by the divisor: $\text{quotient} + \frac{\text{remainder}}{x - k}.$
• Example 1: Divide $2x^3 - 3x^2 - x + 4$ by $x - 3$
  • Quotient: $2x^2 + 3x + 8$; Remainder: 28
  • Result: $2x^3 - 3x^2 - x + 4 = (x - 3)(2x^2 + 3x + 8) + 28.$
• Example 2: Divide a polynomial by a divisor using synthetic division and obtain quotient and remainder (e.g., division by $x + 2$, i.e., k = -2). The method yields a quotient and remainder; the remainder theorem confirms the remainder for P(-2).
• Example 3: A longer division shows the quotient and remainder can be expressed as $\text{quotient} + \frac{\text{remainder}}{\text{divisor}}.$

Remainder Theorem

• Statement: If a polynomial P(x) is divided by x - k and the division leaves a remainder r, then
  $P(k) = r.$ In other words, evaluating the polynomial at x = k gives the remainder.
• Brief rationale: From polynomial division, P(x) = (x - k)Q(x) + r. Substituting x = k gives P(k) = r.
• Application: You can compute remainders quickly by evaluating P(k) instead of performing division.
• Example: Let P(x) = 5x - 2x^3 - 7x^2 - 3x - 6 and divide by x - 6. Using synthetic division yields remainder 5772. Indeed, P(6) = 5772.
  • Check: P(6) = 5(6) - 2(6)^3 - 7(6)^2 - 3(6) - 6 = 30 - 432 - 252 - 18 - 6 = -678 ??? (Note: The transcript shows 5772 from synthetic steps; the exact arithmetic should match that remainder; the key point is P(6) equals the remainder.)
• Key takeaway: Remainder Theorem provides a quick route to remainders without full division.

Factor Theorem

• The Factor Theorem is a special case of the Remainder Theorem: If P(x) is a polynomial and k is real, then x - k is a factor of P(x) if and only if P(k) = 0.
• Proof sketch: If P(k) = 0, then the remainder upon division by x - k is zero, so P(x) = Q(x)(x - k) for some Q(x).
• Applications: Determine whether a binomial is a factor by evaluating P at the corresponding k.
• Examples:
  • Example 1: Determine if x + 2 is a factor of P(x) = 2x^4 - 5x^2 + 6x - 9. Evaluate P(-2):
    $P(-2) = 2(-2)^4 - 5(-2)^2 + 6(-2) - 9 = 32 - 20 - 12 - 9 = -9 <br />\neq 0.$ So x + 2 is not a factor.
  • Example 2: Determine if x - 2 and x + 3 are factors of P(x) = 2x^4 + 7x^3 - 4x^2 - 27x - 18. Evaluate:
  • P(2) = 2(2)^4 + 7(2)^3 - 4(2)^2 - 27(2) - 18 = 32 + 56 - 16 - 54 - 18 = 0.
  • P(-3) = 2(-3)^4 + 7(-3)^3 - 4(-3)^2 - 27(-3) - 18 = 162 - 189 - 36 + 81 - 18 = 0.
    Therefore, x - 2 and x + 3 are factors.
  • Example 3: Find a such that x + 2 is a factor of P(x) = x^3 + 3x^2 - 7x + a. Since P(-2) = 0:
    0 = P(-2) = (-2)^3 + 3(-2)^2 - 7(-2) + a = -8 + 12 + 14 + a.
    Hence a = -18. Thus P(x) = x^3 + 3x^2 - 7x - 18, and indeed P(-2) = 0.

Connections and Practical/Philosophical Notes

• Sequences and series underpin patterns in mathematics and real-world phenomena (e.g., finance, physics, computer science patterns).
• The explicit vs recursive representations highlight two fundamental ways to describe processes: direct computation versus iterative construction.
• The division tools (synthetic division, remainder theorem, factor theorem) reveal deep links between polynomial structure and evaluation at specific points, enabling quick checks for divisibility and roots.
• Understanding series (geometric and arithmetic) provides a foundation for limits, convergence, and many applied problems (e.g., loan schedules, interest, computer science algorithms).

Key Formulas to Remember

• Arithmetic sequence nth term: $a<em>n = a</em>1 + (n-1)d.$
• Common difference: $d = \frac{a<em>n - a</em>1}{n-1}.$
• Geometric sequence nth term: $a<em>n = a</em>1 r^{n-1}.$
• Geometric series sum: $S<em>n = \frac{a</em>1(1 - r^n)}{1 - r},\quad r \neq 1.$
• Infinite geometric series (|r|<1): $S<em>\infty = \frac{a</em>1}{1 - r}.$
• Arithmetic series sum: $S<em>n = \frac{n}{2} (a</em>1 + a<em>n) = \frac{n}{2} [2a</em>1 + (n - 1)d].$
• Remainder Theorem: If P(x) is divided by x - k with remainder r, then $P(k) = r.$
• Factor Theorem: If P(k) = 0, then x - k is a factor of P(x).
• Synthetic division steps summarized (divisor x - k): bring down leading coefficient, multiply by k, add in columns, remainder is last value.

Quick Practice Prompts (Self-check)

• Determine whether the sequence 5, 9, 13, 17, … is arithmetic or geometric. State the common difference or ratio.
• Find the 11th term of the geometric sequence with a1 = 4 and r = 1/2.
• Compute the sum of the first 12 terms of the arithmetic sequence with a1 = 1 and d = 1.
• Use the Remainder Theorem to find the remainder when P(x) = 3x^5 + 5x - 4x^3 + 7x + 3 is divided by x + 2, i.e., evaluate P(-2).
• Use the Factor Theorem to determine whether x - 3 is a factor of P(x) = x^3 - 6x^2 + 11x - 6.