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Lesson 2 of 4

Arithmetic Progressions · Lesson 2 of 4

nth Term of an AP

“It’s the ultimate cheat code for mathematicians who are far too lazy to count by twos all the way to a hundred.”

Learning Objectives

• Understand why the nth-term formula is needed. • Derive and use the formula aₙ = a + (n − 1)d. • Find any required term of an Arithmetic Progression. • Find the position of a given term and check whether a number belongs to an AP. • Solve problems involving missing information, terms from the end and real-life situations.

In the previous lesson, we learned that an Arithmetic Progression is created by repeatedly adding the same common difference. That works perfectly when we need the next few terms. But what if we want the 50th term, the 100th term or even the 1000th term? Writing every term one by one would be slow and unnecessary.

We Need a Shortcut

Instead of building an AP term by term, we can use its first term, common difference and position to jump directly to any term we want.

Finding a Distant Term

Suppose a person's monthly salary is ₹20,000 in the first year and increases by ₹2,000 every year. The salaries form the AP 20,000, 22,000, 24,000, 26,000, ... .

Salary APLaTeX
Here the first term is 20,000 and the common difference is 2,000.
Think About It

Finding the salary in the 4th or 5th year is easy by repeatedly adding ₹2,000. But what if we want the salary in the 25th year? There must be a faster way.

Building the nth-Term Formula

Let the first term of an AP be a and the common difference be d. Instead of memorising a formula immediately, let us see how each term is created.

PositionTermWhat Happened?
1st termaNo d added
2nd terma + dd added once
3rd terma + 2dd added twice
4th terma + 3dd added three times
5th terma + 4dd added four times
Pattern in the TermsLaTeX
Notice that the number multiplying d is always one less than the position of the term.

Why Do We Use n − 1?

We start at the first term a. To reach the second term, we make one jump of size d. To reach the third term, we make two jumps. To reach the nth term, we therefore make exactly n − 1 jumps.

MovementNumber of d's Added
From 1st to 2nd term1 jump
From 1st to 3rd term2 jumps
From 1st to 4th term3 jumps
From 1st to nth termn − 1 jumps
The Meaning of n − 1

The first term is already our starting point. That is why reaching the nth term requires n − 1 changes, not n changes.

Definition
nth Term

The term at position n in an Arithmetic Progression is called its nth term. It is represented by aₙ and is also called the general term of the AP.

nth Term of an Arithmetic ProgressionLaTeX
Here a is the first term, d is the common difference and n is the position of the required term.
SymbolMeaning
aFirst term of the AP
dCommon difference
nPosition of the required term
aₙValue of the nth term

Finding a Particular Term

Worked Example: Find the 10th Term

Problem
Find the 10th term of the AP 2, 7, 12, 17, ... .

  1. 1.The first term is a = 2.
  2. 2.The common difference is d = 7 − 2 = 5.
  3. 3.We need the 10th term, so n = 10.
  4. 4.Use aₙ = a + (n − 1)d.
  5. 5.a₁₀ = 2 + (10 − 1)(5).
  6. 6.a₁₀ = 2 + 45.
  7. 7.a₁₀ = 47.
  8. 8.Therefore, the 10th term is 47.
CalculationLaTeX
We reached the 10th term directly without writing the first nine terms.
Worked Example: Finding a Term When d Is Negative

Problem
Find the 20th term of the AP 50, 46, 42, 38, ... .

  1. 1.The first term is a = 50.
  2. 2.The common difference is d = 46 − 50 = −4.
  3. 3.Here n = 20.
  4. 4.Use aₙ = a + (n − 1)d.
  5. 5.a₂₀ = 50 + (20 − 1)(−4).
  6. 6.a₂₀ = 50 − 76.
  7. 7.a₂₀ = −26.
  8. 8.Therefore, the 20th term is −26.
Be Careful with Negative d

When d is negative, keep it inside brackets while substituting. Writing (−4) makes the sign much easier to handle correctly.

Finding the Position of a Given Term

Sometimes the question works backwards. Instead of asking for the value of a particular term, it gives us a value and asks where that value appears in the AP.

Worked Example: Which Term Is −81?

Problem
Which term of the AP 21, 18, 15, 12, ... is −81?

  1. 1.Here a = 21.
  2. 2.The common difference is d = 18 − 21 = −3.
  3. 3.The required term has value aₙ = −81.
  4. 4.Use aₙ = a + (n − 1)d.
  5. 5.−81 = 21 + (n − 1)(−3).
  6. 6.−81 = 24 − 3n.
  7. 7.−105 = −3n.
  8. 8.n = 35.
  9. 9.Therefore, −81 is the 35th term of the AP.
Finding nLaTeX
What Values Can n Take?

When finding the position of a term, n represents a position: 1st, 2nd, 3rd and so on. Therefore, n must be a positive whole number.

Does a Number Belong to an AP?

The nth-term formula can also tell us whether a particular number appears anywhere in an AP. We assume the number is aₙ and solve for n.

Membership Test

If solving aₙ = a + (n − 1)d gives a positive integer value of n, the number is a term of the AP. If n is fractional, zero or negative, it cannot represent a valid position in the AP.

Worked Example: Is 301 a Term?

Problem
Check whether 301 is a term of the AP 5, 11, 17, 23, ... .

  1. 1.Here a = 5 and d = 11 − 5 = 6.
  2. 2.Assume 301 is the nth term.
  3. 3.So aₙ = 301.
  4. 4.Use aₙ = a + (n − 1)d.
  5. 5.301 = 5 + (n − 1)(6).
  6. 6.301 = 6n − 1.
  7. 7.302 = 6n.
  8. 8.n = 151/3.
  9. 9.151/3 is not a positive integer.
  10. 10.Therefore, 301 is not a term of this AP.
A Very Common Mistake

Do not stop after finding a value of n. Always ask whether that value can actually represent a position in a sequence. Positions cannot be 4.5th or 7.2th.

Finding the Number of Terms

If a finite AP gives us its first term, common difference and last term, we can treat the last term as aₙ and solve for n. The value of n tells us how many terms the AP contains.

Last Term

The last term of a finite AP is often represented by l. If the AP contains n terms, then l = aₙ.

Last Term of a Finite APLaTeX
Worked Example: How Many Terms Are There?

Problem
How many terms are there in the AP 7, 13, 19, ... , 205?

  1. 1.The first term is a = 7.
  2. 2.The common difference is d = 13 − 7 = 6.
  3. 3.The last term is aₙ = 205.
  4. 4.Use aₙ = a + (n − 1)d.
  5. 5.205 = 7 + (n − 1)(6).
  6. 6.198 = 6(n − 1).
  7. 7.33 = n − 1.
  8. 8.n = 34.
  9. 9.Therefore, the AP contains 34 terms.

When Two Terms of an AP Are Known

Sometimes neither a nor d is given directly. Instead, we may know two different terms of the AP. Each known term gives us an equation using aₙ = a + (n − 1)d. Solving the two equations gives a and d.

Worked Example: Determine the AP

Problem
The 3rd term of an AP is 5 and the 7th term is 9. Find the AP.

  1. 1.For the 3rd term, n = 3 and a₃ = 5.
  2. 2.So a + (3 − 1)d = 5.
  3. 3.Therefore, a + 2d = 5. ...(1)
  4. 4.For the 7th term, n = 7 and a₇ = 9.
  5. 5.So a + (7 − 1)d = 9.
  6. 6.Therefore, a + 6d = 9. ...(2)
  7. 7.Subtract equation (1) from equation (2).
  8. 8.4d = 4, so d = 1.
  9. 9.Substitute d = 1 into a + 2d = 5.
  10. 10.a + 2 = 5, so a = 3.
  11. 11.Therefore, the AP is 3, 4, 5, 6, 7, ... .
Using Two Known TermsLaTeX

Finding a Term from the End

In a finite AP, we may be asked for a term counted from the last term rather than from the first. There are two useful ways to handle this.

MethodIdea
Method 1Find the total number of terms first, then convert the position from the end into a position from the beginning.
Method 2Reverse the AP. The last term becomes the first term and the sign of the common difference changes.
Position from the EndLaTeX
If an AP has n terms, the rth term from the end is the (n − r + 1)th term from the beginning.
Worked Example: 11th Term from the End

Problem
Find the 11th term from the end of the AP 10, 7, 4, ... , −62.

  1. 1.Here a = 10 and d = 7 − 10 = −3.
  2. 2.The last term is −62.
  3. 3.First find the total number of terms.
  4. 4.−62 = 10 + (n − 1)(−3).
  5. 5.−72 = −3(n − 1).
  6. 6.n − 1 = 24.
  7. 7.n = 25.
  8. 8.The 11th term from the end is the (25 − 11 + 1)th term from the beginning.
  9. 9.That is the 15th term.
  10. 10.a₁₅ = 10 + (15 − 1)(−3).
  11. 11.a₁₅ = 10 − 42 = −32.
  12. 12.Therefore, the 11th term from the end is −32.
Remember the +1

When converting a position from the end, do not use n − r. Use n − r + 1. For example, in a 25-term AP, the 11th term from the end is the 15th term from the beginning, not the 14th.

Alternative Method: Reverse the AP

You can also reverse a finite AP. For 10, 7, 4, ... , −62, the reversed AP begins −62, −59, −56, ... . The common difference changes from −3 to +3. The required 11th term from the end then becomes simply the 11th term of the reversed AP.

Using the nth Term in Real Life

Many real-life situations become nth-term problems when a quantity changes by the same amount repeatedly. The important skill is recognising the AP hidden inside the words.

StepWhat to Do
1Write the first few values as a sequence.
2Check that the difference is constant.
3Identify a and d.
4Decide what n or aₙ represents in the situation.
5Use aₙ = a + (n − 1)d.
Worked Example: Counting Two-Digit Multiples

Problem
How many two-digit numbers are divisible by 3?

  1. 1.The smallest two-digit multiple of 3 is 12.
  2. 2.The largest two-digit multiple of 3 is 99.
  3. 3.The sequence is 12, 15, 18, ... , 99.
  4. 4.This is an AP with a = 12 and d = 3.
  5. 5.Treat the last term 99 as aₙ.
  6. 6.99 = 12 + (n − 1)(3).
  7. 7.87 = 3(n − 1).
  8. 8.29 = n − 1.
  9. 9.n = 30.
  10. 10.Therefore, there are 30 two-digit numbers divisible by 3.
Worked Example: Simple Interest as an AP

Problem
₹1,000 is invested at 8% simple interest per year. Find the total interest earned at the end of the 30th year.

  1. 1.The simple interest earned after the 1st year is ₹80.
  2. 2.After the 2nd year, the total interest is ₹160.
  3. 3.After the 3rd year, the total interest is ₹240.
  4. 4.So the values 80, 160, 240, ... form an AP.
  5. 5.Here a = 80 and d = 80.
  6. 6.We need a₃₀.
  7. 7.a₃₀ = 80 + (30 − 1)(80).
  8. 8.a₃₀ = 80 + 2320.
  9. 9.a₃₀ = 2400.
  10. 10.Therefore, the total simple interest at the end of 30 years is ₹2,400.
Worked Example: Rows of Plants

Problem
A flower bed has 23 rose plants in the first row, 21 in the second row, 19 in the third row and so on. The last row contains 5 plants. How many rows are there?

  1. 1.The numbers of plants form the AP 23, 21, 19, ... , 5.
  2. 2.Here a = 23.
  3. 3.d = 21 − 23 = −2.
  4. 4.The last term is aₙ = 5.
  5. 5.Use aₙ = a + (n − 1)d.
  6. 6.5 = 23 + (n − 1)(−2).
  7. 7.−18 = −2(n − 1).
  8. 8.9 = n − 1.
  9. 9.n = 10.
  10. 10.Therefore, the flower bed has 10 rows.

Identify the Four Values First

A Simple Problem-Solving Habit

Before substituting anything into the nth-term formula, write down what you know: a = ?, d = ?, n = ?, aₙ = ?. Usually three of these are known and the fourth is what you need to find.

Question TypeUsually KnownUsually Find
Find the 25th terma, d, naₙ
Which term is 75?a, d, aₙn
How many terms end at 120?a, d, last termn
Two terms of the AP are givenTwo equationsa and d
Common Mistakes

• Using nd instead of (n − 1)d. • Finding the common difference in the wrong order. • Losing the negative sign when d is negative. • Confusing n, which is a position, with aₙ, which is the value of a term. • Accepting a fractional or non-positive value of n as a valid term position. • Using n − r instead of n − r + 1 when counting a term from the end.

Quiz

Quick check

Which formula gives the nth term of an Arithmetic Progression?

Quick check

Why does the nth-term formula contain n − 1?

Quick check

What is the 10th term of the AP 2, 7, 12, 17, ... ?

Quick check

When n represents the position of a term in an AP, what kind of value must n be?

Quick check

In an AP with n terms, which term from the beginning is the rth term from the end?

Quick check

How many terms are there in the AP 23, 21, 19, ... , 5?

Practice Problems

Practice Questions
  1. Find the 20th term of the AP 4, 9, 14, 19, ... .
  2. Find the 30th term of the AP 10, 7, 4, 1, ... .
  3. Which term of the AP 3, 8, 13, 18, ... is 78?
  4. Check whether 150 is a term of the AP 6, 11, 16, 21, ... .
  5. Check whether −150 is a term of the AP 11, 8, 5, 2, ... .
  6. Find the number of terms in the AP 7, 13, 19, ... , 205.
  7. Find the AP whose 4th term is 14 and 9th term is 34.
  8. Find the 8th term from the end of the AP 5, 9, 13, ... , 101.
  9. How many multiples of 4 lie between 10 and 250?
  10. How many three-digit numbers are divisible by 7?

Key Takeaways

Key Takeaways

• The nth-term formula allows us to find any term of an AP without writing all the previous terms. • The nth term of an AP is aₙ = a + (n − 1)d. • The expression n − 1 appears because we make n − 1 jumps from the first term to reach the nth term. • We can use the same formula backwards to find the position n of a known term. • A number belongs to an AP only if the value of n obtained is a positive integer. • For a finite AP, the last term l can be written as l = a + (n − 1)d. • Two known terms can give two equations that allow us to find a and d. • The rth term from the end of an n-term AP is the (n − r + 1)th term from the beginning.

Previous · Lesson 1

Arithmetic Progressions

Next · Lesson 3

Sum of First n Terms of an AP