Pre-Calculus - Sequences & Series
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Questions and Answers

What defines coterminal angles?

  • They can only be measured in radians.
  • They have different initial and terminal sides.
  • They are always acute angles.
  • They have the same initial and terminal sides. (correct)

What is the reference angle?

  • An angle greater than 90 degrees.
  • An angle that can be negative.
  • An acute angle always less than 90 degrees. (correct)
  • A negative angle measured from the terminal side.

In the unit circle, what does the coordinate (x, y) correspond to?

  • (r, θ)
  • (θ, r)
  • (sin θ, cos θ)
  • (cos θ, sin θ) (correct)

What is the geometric formula for the sum of the first n terms in a geometric series?

<p>$S_n = \frac{a_1(1 - r^n)}{1 - r}$ (B)</p> Signup and view all the answers

How is an angle measured in degrees converted to radians?

<p>By dividing the degree measure by 180 and then multiplying by π. (C)</p> Signup and view all the answers

What is the terminal side of an angle?

<p>The position after the angle has been rotated. (D)</p> Signup and view all the answers

Which term is the common difference in an arithmetic series?

<p>$d$ (C)</p> Signup and view all the answers

Which of the following correctly describes a circular function?

<p>Functions determined by the length of arcs on a circle. (C)</p> Signup and view all the answers

What is the nature of the angle measured in radians (s)?

<p>It can be either positive or negative. (A)</p> Signup and view all the answers

What is the result of expanding $(3x - 2y)^4$?

<p>$81x^4 - 216x^3y + 216x^2y^2 - 96xy^3 + 16y^4$ (C)</p> Signup and view all the answers

Which of the following describes a negative angle correctly?

<p>An angle measured in a clockwise direction. (A)</p> Signup and view all the answers

In the context of arithmetic sequences, what does the term $a_n$ represent?

<p>The last term in the sequence (B)</p> Signup and view all the answers

In sigma notation, what does the expression $\sum_{i=1}^{n} a_i$ represent?

<p>The sum of the first n terms (C)</p> Signup and view all the answers

What is the value of $S_n$ for an arithmetic series if $d$ is negative?

<p>It can be negative or positive depending on the terms. (C)</p> Signup and view all the answers

What does the variable 'n' represent in the binomial theorem?

<p>The exponent of the binomial (B)</p> Signup and view all the answers

Which of the following statements is true regarding the common ratio $r$ in a geometric series?

<p>$r$ can be any real number except 1. (D)</p> Signup and view all the answers

Which of the following expressions represents the 5th term of the expansion of (2x - y)⁸?

<p>56x⁴y⁴ (A)</p> Signup and view all the answers

In the context of angle measurement, what does one radian signify?

<p>The central angle that subtends an arc equal to the radius of the circle (C)</p> Signup and view all the answers

How do you convert degrees to radians?

<p>Multiply degrees by π/180 (D)</p> Signup and view all the answers

Which term can be identified as 'a' in the binomial theorem for the expression (a + b)?

<p>The first variable term in the expression (D)</p> Signup and view all the answers

For the expression (x - y), what is the degree of the polynomial?

<p>1 (D)</p> Signup and view all the answers

What is the result when converting 180 degrees into radians?

<p>π (A)</p> Signup and view all the answers

In the expansion of a binomial, which option represents the coefficient of the second term generally?

<p>nC1 (C)</p> Signup and view all the answers

What is the formula for the last term of an arithmetic sequence?

<p>𝑎𝑛 = 𝑎1 + (𝑛 - 1)𝑑 (D)</p> Signup and view all the answers

In a geometric sequence, if the first term is 5 and the common ratio is 3, what is the third term?

<p>45 (B)</p> Signup and view all the answers

Which mathematician is associated with Pascal's Triangle?

<p>Blaise Pascal (B)</p> Signup and view all the answers

What is the binomial expansion of (x + y)^4?

<p>x^4 + 4x^3y^2 + 6x^2y^2 + 4xy^3 + y^4 (D)</p> Signup and view all the answers

If the common ratio of a geometric sequence is not equal to 1, what can be concluded?

<p>The ratio between consecutive terms remains constant. (C)</p> Signup and view all the answers

What does Pascal's Triangle provide when expanding binomial expressions?

<p>The coefficients of the expanded terms (B)</p> Signup and view all the answers

What is the result of expanding (2x - 1)^3?

<p>8x^3 - 12x^2 + 6x - 1 (D)</p> Signup and view all the answers

If an arithmetic sequence has a first term of 2 and a common difference of 3, what is the fifth term?

<p>17 (B)</p> Signup and view all the answers

Which of the following describes a series?

<p>An operation of adding terms (B)</p> Signup and view all the answers

What is the formula for the common ratio in a geometric sequence?

<p>𝑟 = 𝑎𝑛 / 𝑎𝑛-1 (D)</p> Signup and view all the answers

Flashcards

Arithmetic Sequence

A sequence where the difference between consecutive terms is constant.

Geometric Sequence

A sequence where each term after the first is found by multiplying the previous term by a constant.

Arithmetic Formula

an = a1 + (n-1)d

Geometric Formula

an = a1 * r^(n-1)

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Binomial Expansion

Algebraic expansion of powers of a binomial.

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Pascal's Triangle

Triangular array of numbers giving coefficients in binomial expansions.

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Sigma Notation

Shorthand for representing a sum of terms in a sequence.

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Sequence

An ordered list of numbers separated by commas

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Series

The sum of a sequence.

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Common Difference

The constant difference between consecutive terms in an arithmetic sequence.

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What is a sequence?

An ordered list of numbers where each number follows a specific pattern.

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What's an arithmetic sequence?

A sequence where the difference between any two consecutive terms is constant (called the common difference).

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What's a geometric sequence?

A sequence where each term is found by multiplying the previous term by a constant (called the common ratio).

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What is the formula for the nth term in an arithmetic sequence?

an = a1 + (n-1)d, where an is the nth term, a1 is the first term, n is the number of terms, and d is the common difference.

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What is the formula for the nth term in a geometric sequence?

an = a1 * r^(n-1), where an is the nth term, a1 is the first term, n is the number of terms, and r is the common ratio.

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What is a series?

The sum of all terms in a sequence.

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What is sigma notation?

A shorthand way to represent a sum of terms in a sequence.

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What is the Binomial Theorem?

A formula used to expand a binomial raised to a power (x + y)^n.

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What is the formula for the rth term in a binomial expansion?

The formula for the rth term in a binomial expansion is given by: 𝒓𝒕𝒉 = 𝒏𝑪𝒓−𝟏 (𝒂)𝒏−𝒓+𝟏 (𝒃)𝒓−𝟏 where 'n' is the exponent, 'a' is the first term, 'b' is the second term, and 'r' represents the term number.

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What is the significance of the coefficient in the binomial theorem?

The coefficient in the binomial theorem, calculated using combinations, determines the numerical multiplier of each term in the expansion. For example, in the expansion of (x + y)^3, the coefficient of the term x^2y is '3' which is determined by 3C1.

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What is the formula for calculating combinations?

The formula for combinations is given by 𝒏𝑪𝒓 = 𝒏! / (𝒓! * (𝒏−𝒓)!), where 'n' represents the total number of items and 'r' represents the number of items chosen.

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What is the meaning of 'n' in the binomial theorem?

'n' represents the exponent of the binomial. It also determines the number of terms that will be present in the expanded form. For example, if 'n' is 3, there will be 4 terms.

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How do you determine the power of the first term ('a') for a given term in the expansion?

The power of the first term 'a' in the rth term is given by (n - r + 1). For example, in the 5th term, the power of 'a' would be (n - 5 + 1) = n - 4.

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How do you determine the power of the second term ('b') for a given term in the expansion?

The power of the second term 'b' in the rth term is given by (r - 1). For example, in the 5th term, the power of 'b' would be (5 - 1) = 4.

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What happens to the powers of 'a' and 'b' in a binomial expansion?

The power of 'a' decreases from 'n' to 0, while the power of 'b' increases from 0 to 'n' as you move from the first term to the last term in the binomial expansion.

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How can you use the binomial theorem to solve problems?

The binomial theorem is used to expand expressions in the form (a + b)^n, where 'n' is a positive integer. It is crucial for solving problems related to probability, combinations, and other mathematical concepts.

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What are coterminal angles?

Coterminal angles are angles that share the same initial and terminal sides. They can be found by adding or subtracting multiples of 360 degrees (or 2π radians) to the original angle.

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What is a reference angle?

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It is always positive and less than 90 degrees.

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What is a unit circle?

A unit circle is a circle with a radius of 1 unit, centered at the origin (0, 0). Each point on the circle represents an angle and its corresponding trigonometric values (cosine and sine).

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What is the relationship between coordinates and trigonometric functions on a unit circle?

For any point (x, y) on the unit circle, the x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of that angle. So, (x, y) = (cos θ, sin θ) where θ is the angle.

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What is radians?

Radians are a unit of angular measurement. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

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Convert degrees to radians

To convert degrees to radians, multiply the angle by 𝜋/180.

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Convert radians to degrees

To convert radians to degrees, multiply the angle by 180/𝜋.

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What is a circular function?

A circular function is a function that relates an angle to a point on the unit circle, using trigonometric ratios like sine, cosine, and tangent.

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Study Notes

Pre-Calculus - Sequences & Series

  • Sequences: Ordered lists of numbers, often separated by commas (...,). Sequences can be finite or infinite.
  • Arithmetic Sequences: Each term is found by adding a constant value (common difference) to the previous term. Formula: An = a₁ + (n − 1)d where:
  • n is the number of terms
  • d is the common difference
  • a₁ is the first term
  • an is the last term
  • Geometric Sequences: Each term is found by multiplying the previous term by a constant value (common ratio). Formula: An = a₁rn-1 where:
  • n is the number of terms
  • r is the common ratio (r ≠ 1)
  • a₁ is the first term
  • an is the last term
  • Series: The sum of the terms in a sequence.
  • Arithmetic Series: Sum Formula: Sn = n/2 (a₁+an) where:
  • n is the number of terms
  • a₁ is the first term
  • an is the last term
  • Geometric Series: Sum Formula: Sn = a₁ (1 - rn) / (1 - r) where:
  • n is the number of terms
  • r is the common ratio (r ≠ 1)
  • a₁ is the first term

Binomial Expansion

  • Binomial Expansion: Expands the power of a binomial (two-term polynomial). Example: (x + y)² = x² + 2xy + y².
  • Pascal's Triangle: A triangular array of numbers where the numbers in each row are coefficients from binomial expansions. The first row contains only 1. The numbers are formed (1) + (1 +1 = 2) + (1+2+1 = 4). Used to determine the coefficients in the expansion of (x + y)n
  • Binomial Theorem: A general formula for expanding (a + b)ⁿ, where: (a + b)ⁿ = Σ(k=0 to n) [n! / (k!(n-k)!) ] * a^(n-k) * b^k
  • n is the exponent
  • k starts from 0
  • a and b are the terms

Angles and Unit Circle

  • Standard Position: Vertex at origin, initial side on positive x-axis.
  • Types of Angles: Acute (0° < a < 90°), right (a = 90°), obtuse (90° < a < 180°), straight (a = 180°).
  • Degree Measure: Measured from 0° to 360°, representing fractions of a circle.
  • Radian Measure: Measured in radians, a complete circle = 2π radians = 360°.
  • Coterminal Angles: Angles that share the same initial and terminal sides.
  • Reference Angle: The acute angle formed by the terminal side and the horizontal axis.
  • Unit Circle: A circle centered at the origin with radius 1. The cosine and sine values of an angle are the x and y coordinates of the point on the unit circle.
  • Trigonometric Functions: Relationships between angles and sides of a right-angled triangle. These include sine, cosine, tangent, and their reciprocals (csc, sec, cot).

Circular Functions

  • Circular Functions: Functions that relate angles to coordinates on a unit circle.
  • Relationships between Trig Functions: Relationships between sine, cosine, tangent and their reciproprocals (cosecant, secant, coltangent). Examples: sin θ = y/r, cos θ = x/r, tan θ = y/x.
  • Trigonometric Function Values in Various Quadrants: Understanding the signs (positive or negative) of sine, cosine, and tangent in each quadrant.
  • Important Note:* This summary assumes access to the included images and diagrams for better understanding of the material.

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Pre-Calculus STEM 11 Q2 PDF

Description

Test your understanding of sequences and series in this Pre-Calculus quiz. Covering both arithmetic and geometric sequences, the quiz explores the formulas for terms and sums. Challenge yourself and solidify your knowledge on these foundational concepts.

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