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Is your relationship worth repairing? Get "Build Before You Promise" → Relationship Repair When Love Is Still There but the Relationship Is Falling Apart Sometimes the relationship does not need another argument. It needs two people willing to stop defending themselves long enough to understand what is actually breaking. By Edwin Ogie • Author of The Relationship Repair Handbook , Build Before You Promise , Emotional Mastery , and The Value of You Not every broken relationship is a relationship without love. Sometimes two people still love each other deeply, but they have become exhausted by unresolved arguments, poor communication, resentment, disappointment and repeated misunderstandings. They may still care. They may still want the relationship. They may e...
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PROGRESSION

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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.

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