AP vs GP Word Problems: Recognise the Sequence From the Wording

AP and GP word problems become much easier when you stop hunting for formulas and start listening to the wording.

Most mistakes happen before the first calculation. A student sees a long story about rows, salaries, deposits, machines, population, or distance, and immediately asks:

Which formula should I use?

The better first question is:

What is happening from one step to the next?

If the same amount is being added or subtracted each time, the pattern is usually an AP.

If the same factor is being multiplied or divided each time, the pattern is usually a GP.

That one difference can save the whole question.

Let us build a clear, exam-friendly way to recognise the sequence from the wording.

First, Know What AP and GP Are Really Saying

An AP, or arithmetic progression, is a list where every term changes by the same difference.

For example:

5, 8, 11, 14, 17, ...

Each term increases by 3.

So this is an AP.

The common difference is:

d = 3

A GP, or geometric progression, is a list where every term changes by the same ratio.

For example:

4, 8, 16, 32, 64, ...

Each term is multiplied by 2.

So this is a GP.

The common ratio is:

r = 2

The difference between them is simple, but word problems hide it inside sentences.

The Main Difference in One Table

Question to ask	AP	GP
What changes each time?	Same amount	Same factor
How do you move to the next term?	Add or subtract	Multiply or divide
Common value	Common difference d	Common ratio r
Typical words	more, less, increases by, decreases by	doubles, triples, halves, percentage change
Example	Salary increases by Rs 500 every year	Amount increases by 8% every year
Formula for nth term	a_n = a + (n - 1)d	a_n = ar^(n - 1)

The Fast Recognition Test

Before using any formula, use this five-step test.

Step	Ask this	What it tells you
1	What is the first term?	Finds a
2	What happens to get the next term?	Finds the pattern
3	Is the same amount added or subtracted?	Think AP
4	Is the same number multiplied or divided?	Think GP
5	Is the question asking for one term or a total?	Choose term formula or sum formula

Many students jump straight to step 5. That is risky. First identify the pattern. Then decide whether the question needs a term or a sum.

AP Clues in Word Problems

AP wording usually describes steady movement.

The quantity changes by the same amount every time.

Common clues include:

Wording clue	Why it suggests AP
”increases by 5 each time”	Same amount is added
”decreases by 3 each day”	Same amount is subtracted
”Rs 500 more every year”	Fixed increase
”each row has 2 more seats than the previous row”	Fixed extra seats
”cost rises by Rs 50 per metre”	Fixed extra cost
”saves Rs 200 every month”	Fixed repeated addition
”houses are numbered consecutively”	Consecutive numbers form an AP

Look carefully at the words “by” and “more”.

If the question says:

The salary increases by Rs 1,000 every year.

That is AP thinking.

The increase is not changing. It is always Rs 1,000.

If the question says:

Each row has 4 more seats than the row before it.

That is also AP thinking.

The number of extra seats is fixed.

GP Clues in Word Problems

GP wording usually describes repeated scaling.

The quantity changes by the same factor every time.

Common clues include:

Wording clue	Why it suggests GP
”doubles every hour”	Multiplied by 2
”triples every round”	Multiplied by 3
”becomes half each time”	Multiplied by 1/2
”falls by 20% every year”	Multiplied by 80/100 each year
”grows by 10% per year”	Multiplied by 110/100 each year
”one fourth of the remaining amount is removed”	Fraction of current amount changes each time
”compound interest”	Interest is calculated on the changing amount

The most important GP clue is this:

The change depends on the current value.

If something grows by 10%, the actual increase is not the same each time. Ten percent of 1,000 is 100, but ten percent of 1,100 is 110. The increase itself changes because the base changes.

That is why percentage growth and percentage fall usually create GP patterns.

The Biggest Trap: “More” vs “Percent More”

These two sentences look similar, but they create different patterns.

Case 1: The number of students increases by 20 every year.
Case 2: The number of students increases by 20% every year.

Case 1 is AP.

If the first year has 100 students, then:

100, 120, 140, 160, ...

The common difference is:

d = 20

Case 2 is GP.

If the first year has 100 students, then:

100, 120, 144, 172.8, ...

The common ratio is:

r = 1.2

The numbers may both start at 100 and 120, but after that they behave differently.

This is why you must not decide from the first two terms alone if the wording gives a percentage. Read the rule.

Term Question or Sum Question?

After identifying AP or GP, decide what the question is asking.

There are two common possibilities.

It may ask for a particular term:

Find the salary in the 10th year.
Find the number of seats in the 15th row.
Find the amount after 5 years.

Or it may ask for a total:

Find the total salary received in 10 years.
Find the total number of seats in 15 rows.
Find the total distance travelled.

This distinction matters because a term formula and a sum formula answer different questions.

Question asks for	You need
One row, one year, one stage, one term	nth term
Total of many rows, years, stages, or terms	sum

AP Formulas You Actually Need

For an AP:

a = first term
d = common difference
n = number of terms

The nth term is:

a_n = a + (n - 1)d

The sum of first n terms is:

S_n = n/2 [2a + (n - 1)d]

If the last term l is known, you can also use:

S_n = n/2 (a + l)

Use AP when the question is built on equal steps.

GP Formulas You Actually Need

For a GP:

a = first term
r = common ratio
n = number of terms

The nth term is:

a_n = ar^(n - 1)

The sum of first n terms is:

S_n = a(r^n - 1) / (r - 1), when r > 1

If the ratio is between 0 and 1, this version is often neater:

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

Both formulas come from the same idea. Choose the one that keeps the answer positive and simple.

Example 1: Rows of Seats in an Auditorium

Read the question:

The first row of an auditorium has 18 seats. Each next row has 3 more seats than the previous row. How many seats are there in the 12th row?

Look at the clue:

3 more seats than the previous row

This is a fixed increase.

So it is an AP.

Now identify:

a = 18
d = 3
n = 12

Use the nth term formula:

a_n = a + (n - 1)d
a_12 = 18 + (12 - 1)3
a_12 = 18 + 33
a_12 = 51

So the 12th row has 51 seats.

Notice that the question asked for one row, not the total number of seats. That is why we used the nth term formula.

Example 2: Total Seats in the Auditorium

Now change the question slightly:

The first row has 18 seats. Each next row has 3 more seats than the previous row. How many seats are there in the first 12 rows altogether?

The pattern is still AP.

But now the question asks:

altogether

So we need the sum.

a = 18
d = 3
n = 12

Use:

S_n = n/2 [2a + (n - 1)d]

So:

S_12 = 12/2 [2(18) + (12 - 1)3]
S_12 = 6 [36 + 33]
S_12 = 6 x 69
S_12 = 414

There are 414 seats in the first 12 rows.

Same story, same pattern, different question. That is why the formula changed.

Example 3: A Number That Doubles

Read the question:

A page starts with 40 views. The number of views doubles every day. How many views will it have on the 6th day?

Look at the clue:

doubles every day

This means multiply by 2 each time.

So it is a GP.

Identify:

a = 40
r = 2
n = 6

Use the nth term formula:

a_n = ar^(n - 1)
a_6 = 40 x 2^(6 - 1)
a_6 = 40 x 2^5
a_6 = 40 x 32
a_6 = 1280

So the 6th day value is 1280 views.

Example 4: Percentage Depreciation

Read the question:

A machine is worth Rs 50,000. Its value falls by 10% every year. Find its value after 3 years.

The clue is:

falls by 10% every year

This is not a fixed fall of Rs 10,000 every year. It is 10% of the current value each year.

So the machine keeps 90% of its value each year.

The common ratio is:

r = 90/100 = 0.9

The starting value is Rs 50,000.

After 3 years:

Value = 50000 x (0.9)^3
Value = 50000 x 0.729
Value = 36450

So the value after 3 years is Rs 36,450.

Example 5: Fixed Saving Every Month

Read the question:

A student saves Rs 500 every month. How much will the student save in 12 months?

This is a fixed addition.

The monthly savings are:

500, 500, 500, 500, ...

This can be treated as an AP with:

a = 500
d = 0
n = 12

The total is:

500 x 12 = 6000

So the total saving is Rs 6,000.

Here, a simple multiplication is enough. Not every AP question needs a long formula.

Example 6: Deposit Doubles Every Month

Now compare it with this:

A student saves Rs 500 in the first month, Rs 1,000 in the second month, Rs 2,000 in the third month, and continues in the same way. How much is saved in the first 6 months?

The deposits are:

500, 1000, 2000, 4000, ...

Each term is multiplied by 2.

So it is a GP.

Here:

a = 500
r = 2
n = 6

The question asks for total saving, so use the GP sum formula:

S_n = a(r^n - 1) / (r - 1)
S_6 = 500(2^6 - 1) / (2 - 1)
S_6 = 500(64 - 1)
S_6 = 500 x 63
S_6 = 31500

The total saving in 6 months is Rs 31,500.

A Simple Way to Build the First Three Terms

If the wording feels confusing, do not panic. Build the first three terms.

For example:

A plant is 20 cm tall. It grows by 4 cm every week.

Terms:

20, 24, 28, ...

Same difference:

24 - 20 = 4
28 - 24 = 4

So it is AP.

Now compare:

A plant's height increases by 20% every week.

If the height begins at 20 cm, then:

20, 24, 28.8, ...

Same ratio:

24/20 = 1.2
28.8/24 = 1.2

So it is GP.

Watch Out for These Mixed Wording Traps

Some questions are designed to test whether you are reading carefully.

Trap 1: The Added Amount Changes

Read this:

A shop sells 10 units on day 1. Each day, it sells 5 more units than the previous day.

This is AP.

The terms are:

10, 15, 20, 25, ...

Now read this:

A shop sells 10 units on day 1. The increase in sales doubles each day.

This is not automatically a GP for total sales.

If the increases are:

5, 10, 20, ...

Then the daily sales become:

10, 15, 25, 45, ...

Those daily sales are neither a simple AP nor a simple GP.

The increases form a GP, but the sales figures themselves do not.

So ask:

What exactly is forming the sequence?

Trap 2: The Word “Total”

Read this:

Each row has 4 more seats than the previous row. Find the total seats in 20 rows.

The rows form an AP.

The question asks for total, so use AP sum.

Do not stop after finding the 20th row.

Trap 3: Initial Amount vs First Term

In real-life style problems, the first term may depend on how the question counts time.

For example:

A machine costs Rs 80,000 today and loses 15% of its value each year. Find the value after 4 years.

The starting value today is Rs 80,000.

After 1 year:

80000 x 0.85

After 4 years:

80000 x (0.85)^4

Here, today is the starting value, not “year 1 after depreciation”.

Words That Usually Point to AP

Use this list as a quick reading guide.

Words in the question	Think
each next row has 2 more	AP
every year salary rises by Rs 1,500	AP
cost increases by Rs 100 each step	AP
length decreases by 5 cm each cut	AP
numbers are consecutive	AP
equally spaced	AP
fixed installment without interest	AP or simple repeated addition

The phrase “fixed amount” is the heart of AP.

Words That Usually Point to GP

Use this list when the question talks about repeated growth or shrinkage.

Words in the question	Think
doubles	GP
triples	GP
becomes half	GP
one third remains	GP
increases by 12%	GP
decreases by 8%	GP
compound interest	GP
population grows at a fixed percentage	GP
each bounce reaches a fixed fraction of the previous height	GP

The phrase “fixed factor” is the heart of GP.

A Final Decision Flow

Use this flow whenever you see an AP vs GP word problem.

1. Identify the quantity being tracked.
2. Write the first term.
3. Read how the next term is made.
4. Same amount added or subtracted? AP.
5. Same factor multiplied or divided? GP.
6. Asked for one term? Use nth term.
7. Asked for total? Use sum.
8. Check whether time starts before or after the first change.

This simple order prevents most confusion.

How to Practise This Topic

Do not practise AP and GP by memorising separate question types only. Practise recognition.

Take ten mixed word problems and do only the first step:

AP, GP, or neither?

Then write one reason:

AP because the increase is Rs 500 each year.
GP because the value falls by 10% each year.
Neither because the added amount itself changes.

After that, solve the questions.

This trains your mind to read the story before touching the formula.

Frequently Asked Questions

How do I know if a word problem is AP or GP?

Ask how one term changes into the next. If the same amount is added or subtracted each time, it is AP. If the same factor is multiplied or divided each time, it is GP.

Does “increases every year” always mean AP?

No. If it increases by a fixed amount, it is usually AP. If it increases by a fixed percentage, it is usually GP.

Does “decreases” always mean AP?

No. A fixed decrease like Rs 500 each year suggests AP. A percentage decrease like 10% each year suggests GP because the decrease depends on the current value.

Why do percentage questions usually become GP?

Because a percentage is taken from the current value each time. As the current value changes, the actual amount of increase or decrease also changes.

What is the easiest way to check if a sequence is AP?

Subtract consecutive terms. If the difference is the same each time, it is AP.

What is the easiest way to check if a sequence is GP?

Divide each term by the previous term. If the ratio is the same each time, it is GP.

What if the question asks for the total?

First identify whether the terms form AP or GP. Then use the sum formula, not just the nth term formula.

Can a question be neither AP nor GP?

Yes. If the difference is not constant and the ratio is not constant, the sequence may be neither. Some questions have increases that form AP or GP, while the main quantity does not.

Should I memorise clue words?

Clue words help, but meaning matters more. “More” may suggest AP, while “percent more” usually suggests GP. Always build the first few terms if you are unsure.

What should I do first in a difficult word problem?

Identify the quantity being tracked and write the first three terms from the wording. Then test differences for AP and ratios for GP.