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CBSE · Class 11 · Mathematics

Geometric Progression and A.M.–G.M. (Ch 8: 8.4 and 8.5)

4 concepts · 32 practice questions with answers · Free to read

Geometric progression and its nth term

From daily life

A ball that rebounds to half its height each time, or an amount that doubles every year: each step multiplies by the same number.

Simple definition

A sequence is a geometric progression (G.P.) if each term after the first is obtained by multiplying the previous term by a fixed number r, the common ratio. With first term a: aₙ = a rⁿ⁻¹.

Formula: aₙ = a rⁿ⁻¹, r = a₂/a₁

Rule 1: Common ratio — Divide any term by the previous term.

  • Easy: 2, 6, 18, 54: r = 3.
  • Medium: 80, 40, 20, 10: r = 1/2.
  • Exam-type: 3, −6, 12, −24: r = −2.

Rule 2: nth term — Put the values in a rⁿ⁻¹.

  • Easy: a = 2, r = 3: a₅ = 2 × 3⁴ = 162.
  • Medium: a = 64, r = 1/2: a₇ = 64 × (1/2)⁶ = 1.
  • Exam-type: a = 5, r = 2: which term is 640? 5 × 2ⁿ⁻¹ = 640 gives 2ⁿ⁻¹ = 128, n = 8.
Remember

To prove three numbers a, b, c are in G.P., check that b² = ac.

Common mistake

Using r = a₂ − a₁ (that is the common difference of an A.P.).

Memory clue

G.P.: Go by Product, Add for A.P.

Spot the difference

A.P.G.P.
Add the same numberMultiply by the same number
d = a₂ − a₁r = a₂ / a₁

Practice questions

Easy
  1. 1Find the common ratio of 4, 12, 36, 108.
  2. 2Find the 6th term of 3, 6, 12, ….
  3. 3Find the 5th term of 81, 27, 9, ….
Medium
  1. 4Which term of 2, 4, 8, … is 512?
  2. 5The 3rd term of a G.P. is 12 and the 6th term is 96. Find a and r.
  3. 6Find the 8th term of 1, −2, 4, ….
Challenging
  1. 7Are 2, 6, 18 and 4, 6, 9 each in G.P.?
  2. 8Find the positive x for which 2, x, 18 are in G.P.
Show answers (try them yourself first)
  1. 13
  2. 296
  3. 31
  4. 42ⁿ = 512, n = 9
  5. 5r³ = 8, r = 2, a = 3
  6. 6-128
  7. 72, 6, 18: r = 3 yes. 4, 6, 9: r = 3/2 yes.
  8. 8x² = 2 × 18 = 36, so x = 6

Sum of a G.P.

From daily life

If you save ₹1 on day 1, ₹2 on day 2, ₹4 on day 3 and keep doubling, how much have you saved in 10 days? A formula adds a G.P. quickly.

Simple definition

The sum of the first n terms of a G.P. with first term a and common ratio r ≠ 1 is Sₙ = a(rⁿ − 1)/(r − 1) (or a(1 − rⁿ)/(1 − r)). If r = 1, Sₙ = na.

Formula: Sₙ = a (rⁿ − 1)/(r − 1)

Rule 1: Sum of n terms — Find a, r and n, then use the formula.

  • Easy: 1 + 2 + 4 + … (10 terms): S = (2¹⁰ − 1)/(2 − 1) = 1023.
  • Medium: 3 + 6 + 12 + … (6 terms): S = 3(2⁶ − 1) = 189.
  • Exam-type: 81 + 27 + 9 + … (5 terms): S = 81(1 − (1/3)⁵)/(1 − 1/3) = 121.

Rule 2: Find n or a missing value — Rearrange the formula.

  • Easy: 1 + 3 + 9 + … + 243 has n = 6 terms (3⁵ = 243).
  • Medium: Sum = 3(3ⁿ − 1)/2 = 363 gives 3ⁿ − 1 = 242, n = 5.
  • Exam-type: If a = 2, r = 3 and Sₙ = 80, then 3ⁿ = 81, n = 4.
Remember

Check the formula with 2 terms: a + ar = a(r² − 1)/(r − 1) = a(r + 1).

Common mistake

Using the formula with r = 1 (it divides by zero).

Memory clue

First × (power − 1) ÷ (ratio − 1).

Practice questions

Easy
  1. 1Find the sum of 1 + 2 + 4 + 8 + 16.
  2. 2Find the sum of 2 + 6 + 18 + 54.
  3. 3Find the sum of the first 5 terms of 16, 8, 4, ….
Medium
  1. 4Find the sum of the first 7 terms of 1, −2, 4, ….
  2. 5How many terms of 1, 3, 9, … give a sum of 364?
  3. 6Find 4 + 12 + 36 + … + 2916. (find n first)
Challenging
  1. 7A tree is 2 m tall. Each year the new growth is 3/4 of last year's (first year growth 1 m). Find the total height after 3 years of growth.
  2. 8If the sum of the first 3 terms of a G.P. is 7 and r = 2, find the first term.
Show answers (try them yourself first)
  1. 131
  2. 280
  3. 331
  4. 443
  5. 5(3ⁿ − 1)/2 = 364, 3ⁿ = 729, n = 6
  6. 64·3ⁿ⁻¹ = 2916 gives 3ⁿ⁻¹ = 729, n = 7; S = 4(3⁷ − 1)/2 = 4372
  7. 72 + 1 + 0.75 + 0.5625 = 4.3125
  8. 8a(2³ − 1) = 7 gives a = 1

Geometric mean

From daily life

Two numbers on a ruler: the number that makes the three steps grow by the same ratio sits between them. That is the geometric mean.

Simple definition

The geometric mean (G.M.) of two positive numbers a and b is √(ab); it is the middle term of the G.P. a, G, b. To insert n geometric means between a and b, the common ratio is r = (b/a)^(1/(n+1)).

Formula: G = √(ab)

Rule 1: G.M. of two numbers — Multiply and take the square root.

  • Easy: G.M. of 4 and 9 = √36 = 6.
  • Medium: G.M. of 2 and 8 = √16 = 4.
  • Exam-type: G.M. of 3 and 27 = √81 = 9.

Rule 2: Insert means — Find r first.

  • Easy: Insert 2 G.M.s between 3 and 81: r³ = 27, r = 3, so 3, 9, 27, 81.
  • Medium: Insert 1 G.M. between 5 and 20: √100 = 10, so 5, 10, 20.
  • Exam-type: Insert 3 G.M.s between 1 and 16: r⁴ = 16, r = 2, so 1, 2, 4, 8, 16.
Remember

The G.M. is always less than or equal to the A.M. (next topic).

Common mistake

Taking the average instead of √(ab).

Memory clue

Multiply, then root.

Practice questions

Easy
  1. 1Find the G.M. of 4 and 16.
  2. 2Find the G.M. of 2 and 50.
  3. 3Insert 2 G.M.s between 2 and 54.
Medium
  1. 4Insert 3 G.M.s between 2 and 32.
  2. 5Find the G.M. of 1/2 and 8.
  3. 6Insert 4 G.M.s between 1 and 243.
Challenging
  1. 7The A.M. of two numbers is 10 and the G.M. is 8. Find the numbers.
  2. 8Show that 6 is the G.M. of 4 and 9 and of 3 and 12.
Show answers (try them yourself first)
  1. 18
  2. 210
  3. 3r³ = 27, r = 3: 2, 6, 18, 54
  4. 4r⁴ = 16, r = 2: 2, 4, 8, 16, 32
  5. 52
  6. 6r⁵ = 243, r = 3: 1, 3, 9, 27, 81, 243
  7. 7a + b = 20, ab = 64: a, b = 16, 4
  8. 84 × 9 = 36 and 3 × 12 = 36, so G.M. = 6 in both

Relation between A.M. and G.M.

From daily life

Of all rectangles with the same perimeter, the square has the largest area. This is a picture of A.M. ≥ G.M.

Simple definition

The arithmetic mean (A.M.) of a and b is (a + b)/2. For positive numbers, A.M. ≥ G.M.: (a + b)/2 ≥ √(ab), with equality only when a = b.

Formula: (a + b)/2 ≥ √(ab)

Rule 1: Compare A.M. and G.M. — Compute both and compare.

  • Easy: a = 4, b = 9: A.M. = 6.5 ≥ G.M. = 6.
  • Medium: a = 5, b = 5: A.M. = G.M. = 5.
  • Exam-type: a = 1, b = 16: A.M. = 8.5 ≥ G.M. = 4.

Rule 2: Use the inequality — Maximise or minimise using the sum or product.

  • Easy: If x + y = 10 (positive), xy ≤ 25, largest when x = y = 5.
  • Medium: If xy = 36 (positive), x + y ≥ 12.
  • Exam-type: For x > 0: x + 1/x ≥ 2.
Remember

A.M. − G.M. = (√a − √b)²/2 ≥ 0, which proves the result.

Common mistake

Using A.M. ≥ G.M. for negative numbers.

Memory clue

Arithmetic is Always at least Geometric.

Practice questions

Easy
  1. 1Find A.M. and G.M. of 2 and 8.
  2. 2Which is larger for 3 and 12: A.M. or G.M.?
  3. 3If x + y = 20 with x, y > 0, find the maximum of xy.
Medium
  1. 4If xy = 49, x, y > 0, find the minimum of x + y.
  2. 5Show that A.M. = G.M. when a = b.
  3. 6The A.M. and G.M. of two numbers are 25 and 24. Find the numbers.
Challenging
  1. 7Find the minimum value of x + 4/x for x > 0.
  2. 8A rectangle has perimeter 40 m. Find the maximum area.
Show answers (try them yourself first)
  1. 1A.M. = 5, G.M. = 4
  2. 2A.M. = 7.5, G.M. = 6, so A.M.
  3. 3100 (x = y = 10)
  4. 414 (x = y = 7)
  5. 5(a + a)/2 = a and √(a·a) = a
  6. 6a + b = 50, ab = 576: a, b = 32, 18
  7. 7x + 4/x ≥ 2√4 = 4, at x = 2
  8. 8l + w = 20, lw ≤ 100 m², when l = w = 10 m