Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Algebra → Sequences and Series ↓


Arithmetic Series


To understand the concept of an arithmetic series, we must first understand what an arithmetic sequence is. An arithmetic sequence is a list of numbers that has a common difference between each successive term. For example, the sequence 2, 5, 8, 11, 14 is an arithmetic sequence because there is a consistent difference of 3 between each term.

Definition of arithmetic series

An arithmetic series is simply the sum of the terms in an arithmetic sequence. If you add up all the terms in the arithmetic sequence described above, 2 + 5 + 8 + 11 + 14, you will get an arithmetic series.

The sum of the first n terms of an arithmetic sequence is called an arithmetic series.

The formula for the sum of an arithmetic series

The formula to calculate the sum S_n of the first n terms in an arithmetic series is:

S_n = n/2 * (a + l)

Or it can also be expressed as follows:

S_n = n/2 * (2a + (n-1)d)

Where:

  • n is the number of terms.
  • a is the first term of the sequence.
  • l is the last term of the sequence.
  • d is the common difference between consecutive terms.

Formula derivation

Let's take a look at these formulas to understand them better:

Step 1: List the arithmetic series

Consider the arithmetic sequence: a, a + d, a + 2d, ..., a + (n-1)d

The sum of the series is:

S_n = a + (a + d) + (a + 2d) + ... + (a + (n-1)d)

Step 2: Write the series in reverse order

Write the sum in reverse order:

S_n = (a + (n-1)d) + (a + (n-2)d) + ... + a

Step 3: Add both the equations

Adding the two expressions for S_n gives:

2S_n = (a + (a + (n-1)d)) + ((a + d) + (a + (n-2)d)) + ... (a + (n-1)d + a)

All bracketed terms simplify to (2a + (n-1)d) :

Since there are n terms, we have:

2S_n = n * (2a + (n-1)d)

So, after dividing by 2, we get the sum:

S_n = n/2 * (2a + (n-1)d)

Examples of arithmetic series

Example 1

Consider the arithmetic sequence 3, 6, 9, 12, ..., 30. Find the sum of this sequence.

Given:

  • First term a = 3
  • Last term l = 30
  • Common difference d = 3

Find the number of terms n using the formula for the last term:

l = a + (n-1)d

Option:

30 = 3 + (n-1)*3

Simplification:

30 = 3 + 3n - 3
30 = 3n

Hence, n = 10

Now find the sum S_n :

S_n = n/2 * (a + l)
S_10 = 10/2 * (3 + 30) = 5 * 33 = 165

The sum of the sequence is 165.

Visual example: Arithmetic series

3 6 9 12 15 18 21 24 27 30

In this visual example, each block represents a term in the series, the number on the block shows the value of the term. The sequence 3, 6, 9, ... 30 is built visually, and the sum of these block values gives us the arithmetic series.

Example 2

Given the sequence 4, 7, 10, 13, ..., 49, find the sum of the series.

Identify the following:

  • First term a = 4
  • Last term l = 49
  • Common difference d = 3

Find n :

49 = 4 + (n-1)*3

Simplification:

49 = 4 + 3n - 3 49 = 3n + 1 48 = 3n n = 16

Now find the sum S_n :

S_16 = 16/2 * (4 + 49) = 8 * 53 = 424

The sum of this series is 424.

Applications of arithmetic series

Arithmetic series have practical applications in many areas. Here are some scenarios where they are used:

  • Business: Calculating total expenses or revenue when there are changes to regular patterns.
  • Construction: Determining the materials needed for the sequential assembly of steps or repetitive elements.
  • Computer science: Optimization of algorithms that process sequential data.
  • Physics: Sum of distances of objects moving with uniform acceleration.

Practice problems

To further strengthen your understanding of arithmetic series, try solving the following problems:

  1. The sequence 1, 4, 7, ..., 22 forms an arithmetic sequence. Find the sum of this sequence.
  2. Find the sum of the first 25 terms of the sequence 10, 15, 20, ...
  3. If the sum of an arithmetic series is 220, the first term is 2, and the common difference is 4, then how many terms will be there in the series?
  4. Find the sum of all odd numbers between 1 to 100.

Conclusion

Understanding arithmetic series in mathematics is very important, as it applies to many real-life scenarios and various fields of study. Mastering the formulas and being able to find the sum of an arithmetic series efficiently is important for problem-solving and analytical work. By practicing and visualizing these sequences and series, you can gain a solid foundation in the concept of arithmetic series.


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Arithmetic Series

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