Fundamentals of Business Mathematics and Statistics · Arithmetic Progression and Geometric Progression

Applications of AP and GP in Business Problems

Q: How do I know whether to use AP or GP in a business problem?

Look at how the quantity changes. If it changes by the same rupee amount each period, use an AP. If it changes by the same percentage each period, use a GP.

Q: Is straight-line depreciation an AP or a GP?

Straight-line depreciation is an AP, because the same amount is deducted each year, so d is negative. Reducing-balance depreciation is a GP, because the same percentage of the opening value is deducted.

Q: Why does the value after n years use rⁿ and not rⁿ⁻¹?

The starting value is term 1. After one year you are at term 2, so after n years you are at term n + 1, which gives A₀ × rⁿ. Use rⁿ⁻¹ only when a is the first period's value.

Q: Do I need to memorise all the formulas for the exam?

Yes, because the paper is a one-hour objective test. Learn the nth term and sum formulas for both AP and GP, and practise them on small numbers.

Updated 10 October 2026 · Fact-checked

Business problems use an AP when a quantity changes by the same amount each period (salary increments, fixed savings rise) and a GP when it changes by the same percentage each period (reducing-balance depreciation, population growth). Find a, d or r, then apply the nth term or sum formula.

Understand Applications of AP and GP in Business

A progression is a list of numbers that follows a fixed rule. In business, the numbers are yearly salaries, monthly savings, asset values or population counts.

In an Arithmetic Progression (AP), you add the same amount every period. A yearly increment of ₹2,000 is a fixed amount, so salaries form an AP. This is linear growth.

In a Geometric Progression (GP), you multiply by the same factor every period. A 10% yearly rise means multiply by 1.10 each year. A 20% fall means multiply by 0.80. This is percentage growth or decay.

The first skill is spotting which one applies. Ask: is the change a fixed rupee amount or a fixed percentage? Fixed amount means AP. Fixed percentage means GP. Straight-line depreciation is an AP with a negative d. Reducing-balance depreciation is a GP with r below 1.

The second skill is reading the language. The value in year 1 is the first term. The value after n years is usually the (n+1)th term, because the starting value counts as term 1. Many marks are lost here.

Key formulas to remember

nth term of AP: Tₙ = a + (n − 1)d; a is the first term, d is the fixed change. d is negative for a fixed decrease.
Sum of n terms of AP: Sₙ = n/2 × [2a + (n − 1)d] = n/2 × (a + l); Use for total salary or total savings over n periods. l is the last term.
nth term of GP: Tₙ = a × rⁿ⁻¹; r = 1 + rate for growth, r = 1 − rate for decay.
Sum of n terms of GP: Sₙ = a(rⁿ − 1) ÷ (r − 1) for r > 1; Sₙ = a(1 − rⁿ) ÷ (1 − r) for r < 1; Valid when r ≠ 1. Use for total of amounts growing by a fixed percentage.
Value after n periods at a fixed percentage: Aₙ = A₀ × rⁿ; A₀ is the starting value, which is term 1. After n periods you are at the (n+1)th term.

How to solve Applications of AP and GP in Business questions

Use this method for any word problem on salary, savings, depreciation or growth.

1Read the story and decide: fixed amount change (AP) or fixed percentage change (GP).
2Write down the first term a. Note whether the question gives the starting value or the first year's value.
3Find d (the fixed amount) or r (1 + rate for growth, 1 − rate for decay).
4Decide what is asked: a single value (use the nth term) or a total (use the sum).
5Work out n carefully. Count terms, not gaps. Check whether the starting value is term 1.
6Substitute into the formula and calculate with simple numbers.
7Check that your answer is sensible: salary rising, asset value falling, total larger than any one term.

Quickest way: Spot, list, match

When to use it: Use it when the question is an MCQ and the options are far apart, or when n is small.

Decide AP or GP in five seconds from the words 'fixed amount' or 'per cent'.
If n is 3 or 4, just list the terms by adding or multiplying. This is faster than a formula.
For an AP sum, use n/2 × (first + last). You often get the last term in one step.
For percentage decay, multiply step by step with round numbers such as 0.8 or 1.1 and stop.
Eliminate options that go the wrong way, such as a higher value after depreciation.

Common mistakes in Applications of AP and GP in Business

Using GP for a fixed-rupee increment, or AP for a percentage rise.
Students skim the question and do not notice whether the change is an amount or a percentage.
Fix: Underline the change in the question. Rupees means AP. Per cent means GP.
Using n instead of n + 1 when the starting value is given.
The starting value is term 1, so the value after n years is term n + 1.
Fix: Write Aₙ = A₀ × rⁿ for values after n periods. Use Tₙ = a × rⁿ⁻¹ only when a is the first period's value.
Taking r = 0.20 for a 20% decrease.
Students confuse the rate with the ratio.
Fix: For a decrease, r = 1 − 0.20 = 0.80. For an increase, r = 1 + rate.
Finding the nth term when the total is asked.
The words 'total earned' or 'total saved' are missed.
Fix: Circle words like total, altogether and in all. These need a sum formula.
Counting the number of terms wrongly when the increment starts later.
Students count years as gaps instead of terms.
Fix: List the first two and last terms with their year numbers, then check n.

Worked examples

Example 1

A clerk's starting monthly salary is ₹20,000. It rises by ₹1,000 at the start of each following year. What is his total salary for 10 years, taking each year's monthly salary as constant for 12 months?

Show the solution

Fixed increment, so it is an AP with a = 20,000 and d = 1,000, counted per month.
Last year's monthly salary: T₁₀ = 20,000 + 9 × 1,000 = 29,000.
Sum of the monthly salaries over the 10 years: S₁₀ = 10/2 × (20,000 + 29,000) = 5 × 49,000 = 2,45,000.
Each monthly salary lasts 12 months, so total = 12 × 2,45,000 = ₹29,40,000.

Answer: ₹29,40,000

Example 2

A machine costing ₹1,00,000 loses 20% of its value at the end of each year, based on the previous year's value. What is its value after 3 years?

Show the solution

A fixed percentage fall means a GP with r = 1 − 0.20 = 0.80.
The cost ₹1,00,000 is the starting value, so use A₃ = A₀ × r³.
r³ = 0.8 × 0.8 × 0.8 = 0.512.
A₃ = 1,00,000 × 0.512 = ₹51,200.

Answer: ₹51,200

Exam tips

Decide AP or GP first. Once that is right, the rest is substitution.
Many questions are built on the n versus n + 1 trap. Check whether the starting value is given.
Use simple r values like 0.9, 0.8, 1.1 and 1.2 and multiply by hand for small n.
With no negative marking, answer every question. Eliminate options that move in the wrong direction, such as a larger value after depreciation.
Practise reading long word problems quickly and writing down a, d or r before touching a formula.

Practice questions from Arithmetic Progression and Geometric Progression

Applications of AP and GP in Business: frequently asked questions

How do I know whether to use AP or GP in a business problem?

Look at how the quantity changes. If it changes by the same rupee amount each period, use an AP. If it changes by the same percentage each period, use a GP.

Is straight-line depreciation an AP or a GP?

Straight-line depreciation is an AP, because the same amount is deducted each year, so d is negative. Reducing-balance depreciation is a GP, because the same percentage of the opening value is deducted.

Why does the value after n years use rⁿ and not rⁿ⁻¹?

The starting value is term 1. After one year you are at term 2, so after n years you are at term n + 1, which gives A₀ × rⁿ. Use rⁿ⁻¹ only when a is the first period's value.

Do I need to memorise all the formulas for the exam?

Yes, because the paper is a one-hour objective test. Learn the nth term and sum formulas for both AP and GP, and practise them on small numbers.