8.17.a Geometric Progression - Concepts and Formulas

1. Definition of Geometric Progression

A Geometric Progression (GP) is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

General form of a GP:

a, ar, ar^2, ar^3, ar^4, ...

Where:

• a = first term (a != 0)
• r = common ratio (r != 0)
• Each term = previous term * r

Examples of GP:

Sequence	First Term (a)	Common Ratio (r)
2, 6, 18, 54, 162, ...	2	3
100, 50, 25, 12.5, ...	100	1/2
3, -6, 12, -24, 48, ...	3	-2
5, 5, 5, 5, 5, ...	5	1
1, 0.1, 0.01, 0.001, ...	1	0.1
-4, 8, -16, 32, -64, ...	-4	-2

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2. Common Ratio

The common ratio r is obtained by dividing any term by its preceding term:

r = a(n) / a(n-1)

For any three consecutive terms:

r = second term / first term = third term / second term

Key observations:

• If r > 1, the GP is increasing (terms grow larger in magnitude)
• If 0 < r < 1, the GP is decreasing (terms approach zero)
• If r < 0, terms alternate in sign
• If r = 1, the GP is a constant sequence
• If r = -1, terms oscillate between a and -a

Example:

Sequence: 4, 12, 36, 108, 324

r = 12/4 = 3
r = 36/12 = 3
r = 108/36 = 3
r = 324/108 = 3

Common ratio = 3

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3. nth Term of a GP (General Term)

The nth term of a GP is given by:

a(n) = a * r^(n-1)

Where:

• a(n) = nth term
• a = first term
• r = common ratio
• n = position of the term

Derivation:

1st term:  a(1) = a = a * r^0 = a * r^(1-1)
2nd term:  a(2) = ar = a * r^1 = a * r^(2-1)
3rd term:  a(3) = ar^2 = a * r^2 = a * r^(3-1)
4th term:  a(4) = ar^3 = a * r^3 = a * r^(4-1)
...
nth term:  a(n) = a * r^(n-1)

Example 1: Find the 8th term of GP: 3, 6, 12, 24, ...

a = 3, r = 6/3 = 2, n = 8

a(8) = 3 * 2^(8-1)
     = 3 * 2^7
     = 3 * 128
     = 384

Example 2: Find the 6th term of GP: 729, 243, 81, 27, ...

a = 729, r = 243/729 = 1/3, n = 6

a(6) = 729 * (1/3)^(6-1)
     = 729 * (1/3)^5
     = 729 * 1/243
     = 3

Example 3: Find the 10th term of GP: 5, -10, 20, -40, ...

a = 5, r = -10/5 = -2, n = 10

a(10) = 5 * (-2)^(10-1)
      = 5 * (-2)^9
      = 5 * (-512)
      = -2560

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4. Sum of First n Terms of a GP

When r != 1:

Formula (when |r| < 1 or when r < 1):

S(n) = a * (1 - r^n) / (1 - r)

Formula (when r > 1):

S(n) = a * (r^n - 1) / (r - 1)

Both formulas are equivalent; choose whichever avoids negative numbers.

When r = 1:

S(n) = n * a

Derivation:

S(n) = a + ar + ar^2 + ar^3 + ... + ar^(n-1)      ... (i)

Multiply both sides by r:
r * S(n) = ar + ar^2 + ar^3 + ... + ar^(n-1) + ar^n   ... (ii)

Subtract (ii) from (i):
S(n) - r * S(n) = a - ar^n
S(n) * (1 - r) = a * (1 - r^n)
S(n) = a * (1 - r^n) / (1 - r)

Example 1: Find the sum of first 6 terms of GP: 2, 6, 18, 54, ...

a = 2, r = 3, n = 6

S(6) = 2 * (3^6 - 1) / (3 - 1)
     = 2 * (729 - 1) / 2
     = 728

Example 2: Find the sum of first 8 terms of GP: 256, 128, 64, 32, ...

a = 256, r = 1/2, n = 8

S(8) = 256 * (1 - (1/2)^8) / (1 - 1/2)
     = 256 * (1 - 1/256) / (1/2)
     = 256 * (255/256) * 2
     = 510

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5. Sum to Infinity of a GP

When |r| < 1 (i.e., -1 < r < 1), the GP converges and has a finite sum:

S(infinity) = a / (1 - r)     where |r| < 1

Why it works: When |r| < 1, as n approaches infinity, r^n approaches 0:

S(n) = a * (1 - r^n) / (1 - r)

As n -> infinity, r^n -> 0

S(infinity) = a * (1 - 0) / (1 - r) = a / (1 - r)

When |r| >= 1: The sum to infinity does not exist (the series diverges).

Example 1: Find the sum to infinity: 8, 4, 2, 1, 1/2, ...

a = 8, r = 1/2

S(infinity) = 8 / (1 - 1/2)
            = 8 / (1/2)
            = 16

Example 2: Find the sum: 1 - 1/3 + 1/9 - 1/27 + ...

a = 1, r = -1/3

S(infinity) = 1 / (1 - (-1/3))
            = 1 / (1 + 1/3)
            = 1 / (4/3)
            = 3/4

Example 3: Express 0.777... as a fraction.

0.777... = 7/10 + 7/100 + 7/1000 + ...

This is a GP with a = 7/10, r = 1/10

S(infinity) = (7/10) / (1 - 1/10)
            = (7/10) / (9/10)
            = 7/9

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6. Properties of Geometric Progression

Property 1: Multiplication/Division

If a, b, c, d, ... are in GP, then:

• ka, kb, kc, kd, ... are also in GP (same ratio r)
• a/k, b/k, c/k, d/k, ... are also in GP (same ratio r)

Property 2: Power Property

If a, b, c, d, ... are in GP, then:

• a^n, b^n, c^n, d^n, ... are also in GP (ratio becomes r^n)
• 1/a, 1/b, 1/c, 1/d, ... are also in GP (ratio becomes 1/r)

Property 3: Product of Equidistant Terms

In a GP, the product of terms equidistant from the beginning and end is constant:

a(1) * a(n) = a(2) * a(n-1) = a(3) * a(n-2) = ... = constant

Property 4: Three Numbers in GP

If a, b, c are in GP, then:

b/a = c/b
b^2 = a * c

Property 5: Selecting Terms in GP

When choosing unknown terms in GP, use symmetric forms:

Number of terms	Choose as	Common ratio
3 terms	a/r, a, ar	r
4 terms	a/r^3, a/r, ar, ar^3	r^2
5 terms	a/r^2, a/r, a, ar, ar^2	r

Property 6: AP-GP Relationship

• If each term of a GP is raised to the same power, the result is a GP
• The logarithms of terms of a GP form an AP:

If a, ar, ar^2, ar^3, ... is a GP, then:
log(a), log(a) + log(r), log(a) + 2log(r), log(a) + 3log(r), ... is an AP

Property 7: Product of n Terms of GP

If the GP has an odd number of terms, the product of all terms equals the middle term raised to the power n:

Product = (middle term)^n

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7. Geometric Mean (GM)

Single Geometric Mean

The Geometric Mean of two positive numbers a and b is:

GM = sqrt(a * b)

For n positive numbers a1, a2, ..., an:

GM = (a1 * a2 * ... * an)^(1/n)

Inserting n Geometric Means Between Two Numbers

To insert n geometric means between a and b:

Total terms in the new GP = n + 2  (including a and b)

Common ratio r = (b/a)^(1/(n+1))

The n geometric means are:
G(1) = a * r
G(2) = a * r^2
G(3) = a * r^3
...
G(n) = a * r^n

Example: Insert 3 geometric means between 2 and 162.

a = 2, b = 162, n = 3

r = (162/2)^(1/(3+1))
  = (81)^(1/4)
  = 3

G(1) = 2 * 3 = 6
G(2) = 2 * 9 = 18
G(3) = 2 * 27 = 54

Complete GP: 2, 6, 18, 54, 162

Product of n Geometric Means

The product of n geometric means inserted between a and b:

Product of n GMs = (a * b)^(n/2)

Relationship Between AM and GM

For two positive numbers a and b:

AM >= GM

(a + b)/2 >= sqrt(a * b)

Equality holds when a = b

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8. AP vs GP Comparison

Property	Arithmetic Progression (AP)	Geometric Progression (GP)
Definition	Constant difference between terms	Constant ratio between terms
General term	a(n) = a + (n-1)d	a(n) = a * r^(n-1)
Common element	Common difference d = a(n) - a(n-1)	Common ratio r = a(n) / a(n-1)
Sum of n terms	S(n) = n/2[2a + (n-1)d]	S(n) = a(r^n - 1)/(r - 1)
Mean	AM = (a+b)/2	GM = sqrt(ab)
Inserting means	d = (b-a)/(n+1)	r = (b/a)^(1/(n+1))
Infinite sum	Diverges (does not exist)	Converges when `
Growth type	Linear growth	Exponential growth
Example	2, 5, 8, 11, 14, ...	2, 6, 18, 54, 162, ...
Graphical pattern	Straight line	Exponential curve
Can terms be 0?	Yes (d = 0)	No (r != 0, a != 0)
Sum of equidistant terms	a(k) + a(n-k+1) = constant	a(k) * a(n-k+1) = constant

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9. Special Types of GP

1. Infinite Recurring Decimals as GP

0.333... = 3/10 + 3/100 + 3/1000 + ... = (3/10)/(1 - 1/10) = 3/9 = 1/3

0.454545... = 45/100 + 45/10000 + ... = (45/100)/(1 - 1/100) = 45/99 = 5/11

2. Arithmetico-Geometric Progression (AGP)

When an AP and GP are multiplied term by term:

AP: 1, 2, 3, 4, ...
GP: 1, r, r^2, r^3, ...
AGP: 1, 2r, 3r^2, 4r^3, ...

Sum to infinity of AGP (for |r| < 1):

S = a/(1-r) + d*r/(1-r)^2

Where a is the first term of the AP and d is its common difference.

3. Sum of GP-like Series

1 + 2 + 4 + 8 + ... + 2^(n-1) = 2^n - 1

1 + 3 + 9 + 27 + ... + 3^(n-1) = (3^n - 1)/2

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10. Summary of All Formulas

Formula	Expression
Common ratio	r = a(n) / a(n-1)
nth term	a(n) = a * r^(n-1)
Sum of n terms (r != 1)	S(n) = a(1 - r^n)/(1 - r) = a(r^n - 1)/(r - 1)
Sum of n terms (r = 1)	S(n) = na
Sum to infinity (	r
Geometric Mean of a, b	GM = sqrt(a * b)
n GMs between a and b	r = (b/a)^(1/(n+1))
Product of n GMs	(ab)^(n/2)
AM-GM inequality	(a+b)/2 >= sqrt(ab)
Condition for GP	b^2 = ac for three terms a, b, c
nth term from end	l / r^(n-1) where l = last term
Log property	If GP, then logs form an AP

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