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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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