1. Identify AP, GP and HP

What is the question?

These questions usually ask:

“Which of the following numbers are in AP/GP/HP?”

Or:

“If a, b, c are in AP/GP/HP, find the missing term.”

Idea

For three numbers, use one simple test.

AP → Middle = average of first and third

     2b = a + c


GP → Square of middle = product of first and third

     b² = ac


HP → Reciprocals are in AP

     2/b = 1/a + 1/c

Example

Question: If 4, x, 10 are in AP, find x.

For AP:

2x = 4 + 10

2x = 14

x = 7

Answer: 7

Remember

AP → ADD
2b = a + c

GP → MULTIPLY
b² = ac

HP → RECIPROCAL
1/a, 1/b, 1/c are in AP

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2. Finding a Missing Term in AP, GP or HP

What is the question?

These questions give two terms and tell you that the three terms are in AP, GP or HP.

Example 1 — AP

Question: 5, x, 17 are in AP. Find x.

2x = 5 + 17

2x = 22

x = 11

Answer: 11

Example 2 — GP

Question: 2, x, 18 are in GP. Find x.

For GP:

x² = 2 × 18

x² = 36

x = 6

Answer: 6

Example 3 — HP

Question: 3, x, 6 are in HP. Find x.

For HP:

2/x = 1/3 + 1/6

2/x = 1/2

x = 4

Answer: 4

Remember

AP → 2b = a + c

GP → b² = ac

HP → 2/b = 1/a + 1/c

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3. Relationship Between AP, GP and HP

What is the question?

These questions ask you to connect the three types of means.

For example:

“If a, b, c are in AP, what is the arithmetic mean?”

Or:

“Find the relation between AM, GM and HM.”

Idea

For two positive numbers a and b:

Arithmetic Mean:

AM = (a + b)/2


Geometric Mean:

GM = √(ab)


Harmonic Mean:

HM = 2ab/(a + b)

An important relationship is:

AM ≥ GM ≥ HM

Example

Question: Find the AM, GM and HM of 4 and 16.

Step 1: AM

AM = (4 + 16)/2
   = 10

Step 2: GM

GM = √(4 × 16)
   = √64
   = 8

Step 3: HM

HM = 2(4)(16)/(4 + 16)

   = 128/20

   = 6.4

Therefore:

AM = 10
GM = 8
HM = 6.4

And:

10 > 8 > 6.4

Remember

For positive numbers: AM ≥ GM ≥ HM

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4. Relation Between AP and HP

What is the question?

These questions usually say something like:

“1/a, 1/b, 1/c are in AP. Show that a, b, c are in HP.”

Or the reverse:

“a, b, c are in HP. What can you say about their reciprocals?”

Idea

HP is simply AP applied to reciprocals.

a, b, c in HP

        ↓ take reciprocals

1/a, 1/b, 1/c in AP

Example

Question: If 1/a, 1/b, 1/c are in AP, prove that a, b, c are in HP.

Since they are in AP:

2/b = 1/a + 1/c

Therefore:

2/b = (a + c)/ac

Cross multiply:

2ac = b(a + c)

So:

2/b = 1/a + 1/c

Hence:

a, b, c are in HP.

Remember

HP = AP of reciprocals.

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5. Arithmetic Mean and Geometric Mean Relationship

What is the question?

These questions often ask:

“If the AM and GM of two numbers are given, find the numbers.”

Or:

“If two numbers have AM = X and GM = Y, find their sum/product.”

Idea

For two numbers a and b:

AM = (a + b)/2
GM = √ab

Therefore:

a + b = 2AM

ab = GM²

Example

Question: The AM of two numbers is 5 and their GM is 4. Find the numbers.

From AM:

(a + b)/2 = 5

a + b = 10

From GM:

√ab = 4

ab = 16

So a and b are roots of:

x² - 10x + 16 = 0

Factor:

(x - 2)(x - 8) = 0

Therefore:

a = 2
b = 8

Answer: 2 and 8

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6. AP, GP and HP of the Same Three Numbers

What is the question?

These questions ask:

“Three numbers are simultaneously in AP and GP. Find their relationship.”

Idea

If the same three numbers are in both AP and GP, they must be equal.

For AP:

2b = a + c

For GP:

b² = ac

Since:

b = (a + c)/2

Substitute into GP:

((a + c)/2)² = ac

This gives:

(a - c)² = 0

Therefore:

a = c

And because:

2b = a + c

we get:

a = b = c

Example

Question: If a, b, c are both in AP and GP, prove that a = b = c.

AP → 2b = a + c

GP → b² = ac

Therefore:

((a + c)/2)² = ac

(a - c)² = 0

a = c

Therefore:

a = b = c

Answer: All three numbers are equal.

Remember

Same three terms in AP + GP → all terms are equal.

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Advanced Variants

The following are less basic but are useful for tougher aptitude questions.

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7. Inserting Means Between Two Numbers

What is the question?

Questions may ask:

“Insert 3 arithmetic means between 5 and 21.”

Or:

“Find the geometric means between two numbers.”

Idea

For AP, create an arithmetic sequence.

Example — AP

Question: Insert 3 arithmetic means between 5 and 21.

There will be:

5, _, _, _, 21

Total terms:

3 + 2 = 5

Use:

a₅ = a + 4d

So:

21 = 5 + 4d

16 = 4d

d = 4

Therefore:

5, 9, 13, 17, 21

The three means are:

9, 13, 17

For GP

If inserting n geometric means:

a, G₁, G₂, ..., b

Use:

b = ar^(n+1)

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8. AP and GP Conditions Combined

What is the question?

These questions give different conditions involving AP and GP together.

For example:

“a, b, c are in AP and their product is known. Find the terms.”

Idea

Use the AP relationship first:

a + c = 2b

Then use the other given condition.

Example

Question: Three numbers are in AP. Their sum is 18 and their product is 120. Find the numbers.

Since they are in AP, write them as:

6 - d, 6, 6 + d

Why?

Their middle term is:

18/3 = 6

Product:

(6-d)(6)(6+d) = 120

Using:

(6-d)(6+d) = 36-d²

we get:

6(36-d²) = 120

216 - 6d² = 120

6d² = 96

d² = 16

d = 4

Therefore:

2, 6, 10

Answer: 2, 6, 10

Remember

For 3 AP terms, a very useful form is:

a - d, a, a + d

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9. AP → GP → HP Relationships

What is the question?

These questions give one progression and ask you to construct another.

For example:

“If a, b, c are in AP, find the numbers that are in GP using these terms.”

Or:

“If three numbers are in GP, determine their AM/GM/HP.”

Key idea

For three positive numbers:

AP:
2b = a + c

GP:
b² = ac

HP:
2/b = 1/a + 1/c