Featured post

Understanding Human Behaviour Without Spoken Words

By
Image
Understanding Human Behaviour Without Spoken Words — Edwin Ogie Library Understanding Human Behaviour Without Spoken Words Nonverbal Communication as a core human skill — simple, practical, and classroom-friendly. Chapter Objectives Introduction Meaning & Scope Major Channels Interpreting Behaviour Culture & Ethics Practical Applications Case Illustrations Reflection & Practice Summary & Terms By Edwin Ogie Library — clear, usable lessons for students and teachers. Chapter Objectives At the end of this chapter, the reader should be able to: Clearly define nonverbal communication and explain its role in human interaction. Identify and interpret major forms of nonverbal behaviour with accuracy. Analyse behaviour using clusters of cues rather than isolated signals. Apply nonverbal awareness eff...
Post a Comment

PROGRESSION

By
Lesson Note: Progressions (nth Term & Sum of A.P. and G.P.)

Progression: nth Term of a Progression

This lesson note explains how to determine the nth term in a progression. We consider both:

A. Arithmetic Progression (A.P.)

The nth term of an A.P. is given by:

an = a + (n - 1)d, where:

  • a is the first term,
  • d is the common difference, and
  • n is the term number.
  • Example 1: For a = 2 and d = 3, find the 5th term.
    a5 = 2 + (5 - 1)×3 = 2 + 12 = 14.
  • Example 2: For a = 7 and d = 2, find the 8th term.
    a8 = 7 + (8 - 1)×2 = 7 + 14 = 21.
  • Example 3: For a = 1 and d = 4, find the 6th term.
    a6 = 1 + (6 - 1)×4 = 1 + 20 = 21.
  • Example 4: For a = 10 and d = -2, find the 4th term.
    a4 = 10 + (4 - 1)×(-2) = 10 - 6 = 4.

B. Geometric Progression (G.P.)

The nth term of a G.P. is given by:

an = a × r(n - 1), where:

  • a is the first term,
  • r is the common ratio, and
  • n is the term number.
  • Example 1: For a = 2 and r = 2, find the 5th term.
    a5 = 2 × 2(5 - 1) = 2 × 16 = 32.
  • Example 2: For a = 3 and r = 3, find the 4th term.
    a4 = 3 × 3(4 - 1) = 3 × 27 = 81.
  • Example 3: For a = 5 and r = 0.5, find the 6th term.
    a6 = 5 × (0.5)(6 - 1) = 5 × 0.03125 ≈ 0.15625.
  • Example 4: For a = 10 and r = 0.8, find the 3rd term.
    a3 = 10 × (0.8)(3 - 1) = 10 × 0.64 = 6.4.

Progression: Sum of A.P. and G.P.

This section explains how to find the sum of the first n terms in both Arithmetic and Geometric progressions.

A. Sum of an Arithmetic Progression (A.P.)

The sum of the first n terms of an A.P. is given by:

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

  • Example 1: For a = 2 and d = 3 (5 terms),
    S5 = 5/2 [2×2 + (5 - 1)×3] = 5/2 × 16 = 40.
  • Example 2: For a = 7 and d = 2 (8 terms),
    S8 = 8/2 [2×7 + (8 - 1)×2] = 4 × 28 = 112.
  • Example 3: For a = 1 and d = 4 (6 terms),
    S6 = 6/2 [2×1 + (6 - 1)×4] = 3 × 22 = 66.
  • Example 4: For a = 10 and d = -2 (4 terms),
    S4 = 4/2 [2×10 + (4 - 1)×(-2)] = 2 × 14 = 28.

B. Sum of a Geometric Progression (G.P.)

The sum of the first n terms of a G.P. (for r ≠ 1) is given by:

Sn = a (rn - 1) / (r - 1) (if r > 1) or Sn = a (1 - rn) / (1 - r) (if r < 1).

  • Example 1: For a = 2 and r = 2 (5 terms),
    S5 = 2 × (25 - 1)/(2 - 1) = 2 × 31 = 62.
  • Example 2: For a = 3 and r = 3 (4 terms),
    S4 = 3 × (34 - 1)/(3 - 1) = 3 × 40 = 120.
  • Example 3: For a = 5 and r = 0.5 (6 terms),
    S6 = 5 × (1 - 0.56)/(1 - 0.5) ≈ 9.84.
  • Example 4: For a = 10 and r = 0.8 (3 terms),
    S3 = 10 × (1 - 0.83)/(1 - 0.8) = 24.4.

Quiz on Progressions (Multiple Choice)

Total time: 300 seconds

This topic explains how to determine the nth term of a progression, including both arithmetic and geometric progressions. A.P. Example: a = 2, d = 3, n = 5, a5 = 2 + (5-1)*3 = 14. A.P. Example: a = 7, d = 2, n = 8, a8 = 7 + (8-1)*2 = 21. G.P. Example: a = 2, r = 2, n = 5, a5 = 2 × 2^(5-1) = 32. G.P. Example: a = 3, r = 3, n = 4, a4 = 3 × 3^(4-1) = 81. This topic explains how to calculate the sum of terms in arithmetic and geometric progressions. A.P. Example: a = 2, d = 3, n = 5, S5 = 5/2 [2×2 + (5-1)*3] = 40. A.P. Example: a = 7, d = 2, n = 8, S8 = 8/2 [2×7 + (8-1)*2] = 112. G.P. Example: a = 2, r = 2, n = 5, S5 = 2*(2^5-1)/(2-1) = 62. G.P. Example: a = 3, r = 3, n = 4, S4 = 3*(3^4-1)/(3-1) = 120.

Comments

Post a Comment

We’d love to hear from you! Share your thoughts or questions below. Please keep comments positive and meaningful, Comments are welcome — we moderate for spam and civility; please be respectful.

Popular Posts

FORGIVENESS THE SECRET TO A SUCCESSFUL RELATIONSHIP

By
Image
FORGIVENESS THE SECRET TO A SUCCESSFUL RELATIONSHIP Forgiveness is a critical aspect of human relationships, particularly in marriage. Treating others with mercy and leniency requires you to overlook misunderstandings and the wrongs they have done to you. The capacity to forgive is a fundamental virtue that is the linchpin of safeguarding relationships and leading a flourishing life. Lack of forgiveness causes lots of damage to marriages, and it is often the root cause of divorce in our society. It is, therefore, vital to understand that forgiveness is the key to a lasting, successful, and fulfilling marriage. Before embarking on the journey of marriage, individuals should take responsibility and learn to know themselves deeply. It is essential to realize that marriage is not a walk in the park. It requires perseverance, tenacity, and a loving spirit from the partners involved. When hurt arises, forgiveness is an indispensable tool that partners must employ to calm their difference...
Read more

Mastering the Art of Present Steps for Future Triumphs

By
Mastering the Art of Present Steps for Future Triumphs" In the pursuit of victory, every step forward is pivotal. It's essential to navigate your present position with care, for it is the springboard to your future achievements. How you handle your current endeavors directly impacts the path to your next conquests. Attempting to bypass or disregard your present circumstances can undermine your success in the steps that follow. In the realm of business, strategic maneuvering is essential to attaining your desired goals. Step into Consciousness: Harnessing Experience for Future Benefits Experience serves as our greatest teacher, guiding us towards tomorrow's rewards. Often, we fall into the trap of assuming the world revolves around us, treating our current situations with indifference and overlooking their implications for future outcomes. It's crucial to be mindful of how we approach our present steps, recognizing their significance in shaping tomorrow's victories....
Read more

Navigating Life's Complexities Through Self-Consciousness

By
EO Navigating Life’s Complexities Through Self-Consciousness A practical, story-driven guide to building mental resilience, clarity and better relationships in daily life. Expand all Collapse all Print Download .doc Introduction What is Self-Consciousness? Why it Matters The Nigerian Context Stories in Action Root Causes Upbringing & Environment Strategies to Grow Applications Exercises & Plans Scripts & Conversation Frames Resources Conclusion Introduction In a world of constant noise, stress, and social pressure, developing self-consciousness — the steady ability to notice yourself and choose — is essential. This post explains what it is, why it matters (especially in Nigeria), and how to build it with concrete stories...
Read more