Sequences & Progressions Tutorial | EverExams.com

Sequences & Progressions

What is a Sequence?

A sequence is a special type of function from a subset of natural numbers to a set of numbers. It’s an ordered list of numbers following a specific pattern.

Notation: A sequence is denoted by {aₙ} or (aₙ) where aₙ is the nth term.

Example: aₙ = 2n + 1 generates the sequence: 3, 5, 7, 9, …
Quick Tip

Think of a sequence as an ordered list. The position of each term matters!

Types of Sequences

Finite Sequence: Has a limited number of terms (domain is finite).

Infinite Sequence: Continues indefinitely with no last term.

Real Sequence: All terms are real numbers.

Progression: A sequence where terms increase or decrease in a specific pattern.

Memory Aid

Remember: All progressions are sequences, but not all sequences are progressions!

Types of Progressions

1. Arithmetic Progression (A.P.): Difference between consecutive terms is constant.

2. Geometric Progression (G.P.): Ratio between consecutive terms is constant.

3. Harmonic Progression (H.P.): Reciprocals of terms form an A.P.

Arithmetic Progression (A.P.)

Definition & Properties

A sequence {aₙ} is an A.P. if aₙ₊₁ – aₙ = constant for all n. This constant is called the common difference (d).

Standard form: a, a+d, a+2d, a+3d, …

Important Notes:

  • Common difference (d) cannot be zero in an A.P.
  • Zero can be a term in an A.P., but not the common difference.
Quick Check

To verify if a sequence is A.P., subtract consecutive terms. If you get the same value each time, it’s an A.P.

Formulas for A.P.

nth term: aₙ = a₁ + (n-1)d

aₙ = a + (n-1)d

Sum of first n terms: Sₙ = n/2 [2a + (n-1)d] or Sₙ = n/2 (a + aₙ)

Sₙ = n/2 [2a + (n-1)d] = n/2 (first term + last term)

Arithmetic Mean (A.M.): The A.M. between a and b is (a+b)/2

If a, A, b are in A.P., then A = (a+b)/2
Memory Trick

For A.P. formulas: “First plus last, times n, divided by two” for sum. “Start plus (n-1) times d” for nth term.

Geometric Progression (G.P.)

Definition & Properties

A sequence {aₙ} is a G.P. if aₙ₊₁ / aₙ = constant for all n. This constant is called the common ratio (r).

Standard form: a, ar, ar², ar³, …

Important Notes:

  • Common ratio (r) cannot be zero in a G.P.
  • No term of a G.P. can be zero.
  • If r > 1, terms increase; if 0 < r < 1, terms decrease; if r < 0, terms alternate signs.
Quick Check

To verify if a sequence is G.P., divide consecutive terms. If you get the same value each time, it’s a G.P.

Formulas for G.P.

nth term: aₙ = a₁ × rⁿ⁻¹

aₙ = arⁿ⁻¹

Sum of first n terms:

Sₙ = a(1 – rⁿ)/(1 – r) when r ≠ 1
Sₙ = na when r = 1

Sum of infinite G.P.: Only converges when |r| < 1

S∞ = a/(1 – r) for |r| < 1

Geometric Mean (G.M.): The G.M. between a and b is √(ab)

If a, G, b are in G.P., then G = √(ab)
Memory Trick

For G.P. sum: “a times (1 minus r to the n) over (1 minus r)” for finite sum, and “a over (1 minus r)” for infinite sum when |r| < 1.

Harmonic Progression (H.P.)

Definition & Properties

A sequence is called a Harmonic Progression if the reciprocals of its terms form an Arithmetic Progression.

Standard form: 1/a, 1/(a+d), 1/(a+2d), 1/(a+3d), …

Important Notes:

  • No term of an H.P. can be zero.
  • There’s no general formula for the sum of n terms of an H.P.
  • All results of A.P. can be used for H.P. by taking reciprocals.
Quick Tip

When dealing with H.P., always convert to A.P. by taking reciprocals, solve the problem in A.P., then convert back.

Harmonic Mean & Relations

Harmonic Mean (H.M.): The H.M. between a and b is 2ab/(a+b)

If a, H, b are in H.P., then H = 2ab/(a+b)

Relation between A.M., G.M., and H.M.:

For two positive numbers a and b: A ≥ G ≥ H
G² = A × H
A = (a+b)/2, G = √(ab), H = 2ab/(a+b)
Memory Aid

Remember the inequality: A ≥ G ≥ H for positive numbers. “Always Greater than Geometric, Greater than Harmonic.”

Practice Quiz

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Study Guidelines

How to Master Sequences & Progressions

  1. Understand the basics: Start with clear definitions of sequences, series, and progressions.
  2. Memorize key formulas: Write down all A.P., G.P., and H.P. formulas and practice them regularly.
  3. Practice identification: Learn to quickly identify which type of progression a given sequence belongs to.
  4. Solve step-by-step: Don’t skip steps when solving problems. Show your work to understand the process.
  5. Work on real-world problems: Apply sequences and progressions to real-life scenarios like population growth, financial calculations, etc.
  6. Take timed practice tests: Simulate exam conditions to improve speed and accuracy.
  7. Review mistakes: Analyze errors in practice problems to avoid repeating them.
  8. Use visual aids: Draw diagrams or graphs to visualize sequences and their patterns.
  9. Connect concepts: Understand how A.P., G.P., and H.P. are related through reciprocals and means.
  10. Teach someone else: Explaining concepts to others is one of the best ways to solidify your understanding.

Common Mistakes to Avoid

  • Confusing sequences with series (sequence is the list, series is the sum)
  • Using A.P. formulas for G.P. problems and vice versa
  • Forgetting that common difference cannot be zero in A.P.
  • Applying infinite G.P. sum formula when |r| ≥ 1
  • Not checking if terms are positive when using A ≥ G ≥ H inequality
  • Miscalculating the number of terms in a progression
  • Forgetting to convert H.P. to A.P. by taking reciprocals

Exam Tips

  • Read carefully: Pay attention to whether the problem asks for a sequence or series.
  • Identify first: Determine the type of progression before applying formulas.
  • Check conditions: Verify conditions like r < 1 for infinite G.P. sum.
  • Estimate: Use estimation to check if your answer is reasonable.
  • Manage time: Allocate time based on marks – don’t spend too long on one question.
  • Show formulas: Even if you make a calculation error, showing correct formulas may earn partial credit.