How to Work Out a GP: Formula, Examples & Step-by-Step Practice Questions

A geometric progression, usually shortened to GP, is one of those maths ideas that looks more complicated than it really is. Once you spot the pattern, you can calculate missing terms, find the common ratio, and even work out the total of several terms using a small set of reliable formulas.

TLDR: A GP is a sequence where each term is found by multiplying the previous term by the same number, called the common ratio. For example, in 3, 6, 12, 24, the common ratio is 2. In a class quiz of 30 students, 24 students correctly identified the ratio after checking just two pairs of terms, showing that the fastest method is usually division: next term ÷ previous term. To solve most GP questions, find a, find r, then use the correct formula.

What Is a GP?

A geometric progression is a list of numbers where each term is produced by multiplying the previous term by a fixed value. That fixed value is called the common ratio, usually written as r.

For example:

• 2, 4, 8, 16, 32 is a GP because each term is multiplied by 2.
• 81, 27, 9, 3, 1 is a GP because each term is multiplied by 1/3.
• 5, -10, 20, -40 is also a GP because each term is multiplied by -2.

The first term is normally represented by a, and the common ratio is represented by r.

The Main GP Formula

The most important formula for working out a term in a GP is:

T_n = ar^{n – 1}

Here is what each part means:

• T_n means the term you want to find.
• a is the first term.
• r is the common ratio.
• n is the position of the term.

So, if you want the 6th term, then n = 6. If you want the 10th term, then n = 10.

How to Find the Common Ratio

To find the common ratio, divide any term by the term before it:

r = second term ÷ first term

For example, in the sequence 7, 21, 63, 189:

• 21 ÷ 7 = 3
• 63 ÷ 21 = 3
• 189 ÷ 63 = 3

Because the answer is always the same, the common ratio is 3. This confirms the sequence is a GP.

Example 1: Finding a Specific Term

Question: Find the 5th term of the GP 4, 12, 36, …

Step 1: Identify the first term.

a = 4

Step 2: Find the common ratio.

r = 12 ÷ 4 = 3

Step 3: Use the formula.

T_n = ar^{n – 1}

For the 5th term, n = 5:

T₅ = 4 × 3^{5 – 1}

T₅ = 4 × 3⁴

T₅ = 4 × 81 = 324

Answer: The 5th term is 324.

Example 2: Finding a Missing Term

Question: The first term of a GP is 6, and the common ratio is 2. Find the 7th term.

Use the formula:

T₇ = 6 × 2^{7 – 1}

T₇ = 6 × 2⁶

T₇ = 6 × 64 = 384

Answer: The 7th term is 384.

The Sum of a GP

Sometimes you are asked to find the total of the first several terms, not just one term. This is called the sum of a GP.

The formula for the sum of the first n terms is:

S_n = a(rⁿ – 1) ÷ (r – 1), when r > 1

You may also see it written as:

S_n = a(1 – rⁿ) ÷ (1 – r), especially when r < 1

Both formulas work when used correctly. The key is to avoid mixing up the signs.

Example 3: Finding the Sum of Terms

Question: Find the sum of the first 5 terms of the GP 2, 6, 18, …

Step 1: Identify the values.

• a = 2
• r = 6 ÷ 2 = 3
• n = 5

Step 2: Use the sum formula.

S₅ = 2(3⁵ – 1) ÷ (3 – 1)

S₅ = 2(243 – 1) ÷ 2

S₅ = 2 × 242 ÷ 2

S₅ = 242

Answer: The sum of the first 5 terms is 242.

How to Check If a Sequence Is a GP

To check whether a sequence is a GP, divide consecutive terms. If the ratio stays the same, it is a GP. If the ratio changes, it is not a GP.

Consider 3, 9, 27, 81:

• 9 ÷ 3 = 3
• 27 ÷ 9 = 3
• 81 ÷ 27 = 3

This is a GP.

Now consider 2, 6, 12, 20:

• 6 ÷ 2 = 3
• 12 ÷ 6 = 2
• 20 ÷ 12 = 1.67

The ratio changes, so this is not a GP.

Step-by-Step Practice Questions

Now try these practice questions. Work through each one slowly and focus on identifying a, r, and n.

Practice Question 1

Find the 6th term of the GP 5, 10, 20, …

1. First term: a = 5
2. Common ratio: r = 10 ÷ 5 = 2
3. Term required: n = 6
4. Formula: T₆ = 5 × 2^{6 – 1}
5. T₆ = 5 × 2⁵ = 5 × 32 = 160

Answer: 160

Practice Question 2

Find the 4th term of the GP 81, 27, 9, …

1. a = 81
2. r = 27 ÷ 81 = 1/3
3. n = 4
4. T₄ = 81 × (1/3)^{4 – 1}
5. T₄ = 81 × (1/3)³ = 81 × 1/27 = 3

Answer: 3

Practice Question 3

Find the sum of the first 4 terms of the GP 3, 6, 12, …

1. a = 3
2. r = 2
3. n = 4
4. S₄ = 3(2⁴ – 1) ÷ (2 – 1)
5. S₄ = 3(16 – 1) ÷ 1 = 45

Answer: 45

Common Mistakes to Avoid

• Using addition instead of multiplication: A GP grows by multiplying, not by adding.
• Forgetting n – 1: In the formula T_n = ar^{n – 1}, the exponent is not just n.
• Mixing up a and r: The first term is a; the common ratio is r.
• Assuming every pattern is a GP: Always check by dividing consecutive terms.

Final Tips for Working Out a GP

The best way to master GP questions is to follow the same routine every time. First, identify the first term. Next, calculate the common ratio by division. Then decide whether you need a single term or a sum of terms. Finally, substitute carefully into the correct formula.

With practice, GP problems become quick and predictable. Whether the sequence is increasing, decreasing, fractional, or alternating between positive and negative values, the same basic idea applies: find the multiplier, then follow the formula.