Skip to lesson content

Lesson 4 of 4

Arithmetic Progressions · Lesson 4 of 4

Chapter Summary and Practice

“Common differences, nth terms and sums line up neatly for one final sequence challenge.”

Learning Objectives

• Revise the essential ideas and formulas of Arithmetic Progressions. • Decide whether a problem requires the common difference, nth-term formula or sum formula. • Recognise and avoid common mistakes in AP problems. • Apply the complete chapter to mixed, exam-style and real-life problems.

You have now learned the three main ideas of Arithmetic Progressions: how to recognise an AP, how to find any particular term, and how to find the sum of several terms. This lesson brings those ideas together so that you can decide what to do when a question does not directly tell you which formula to use.

The Big Picture

Most questions in this chapter come down to one decision: Are you studying the pattern between terms, finding one particular term, or finding the total of many terms?

Arithmetic Progressions at a Glance

Definition
Arithmetic Progression

An Arithmetic Progression is a sequence in which the difference between every pair of consecutive terms is constant.

General Form of an APLaTeX
Here a is the first term and d is the common difference.
Common DifferenceLaTeX
Always subtract the earlier term from the term immediately after it.
Remember

The common difference d may be positive, negative or zero. A positive d gives an increasing AP, a negative d gives a decreasing AP, and d = 0 gives a constant AP.

How to Check Whether a Sequence Is an AP

Calculate the differences between consecutive terms. If every difference is equal, the sequence is an AP. If even one of the differences is different, the sequence is not an AP.

AP TestLaTeX

Finding One Particular Term

nth Term FormulaLaTeX
Use this when the question asks about one particular position or term.
SymbolMeaning
aFirst term
dCommon difference
nPosition of the term
aₙValue of the term at position n
Why n − 1?

The formula contains n − 1 because we begin at the first term. To reach the nth term, we make exactly n − 1 jumps of size d.

Last TermLaTeX
For a finite AP, the nth term is also the last term l.

Finding the Total of Several Terms

Sum of First n TermsLaTeX
Use this form when the first term, common difference and number of terms are known.
Sum Using the Last TermLaTeX
Use this form when the first term, last term and number of terms are known.
Finding a Term from SumsLaTeX
Subtracting the sum of the first n − 1 terms from the sum of the first n terms leaves only the nth term.
Sum of First n Positive IntegersLaTeX
The numbers 1, 2, 3, ... themselves form an AP.

Arithmetic Mean

If three numbers a, b and c are consecutive terms of an AP, then the middle number lies exactly halfway between the other two.

Arithmetic MeanLaTeX
The middle term b is called the arithmetic mean of a and c.

Formula Sheet

IdeaFormula
Common differenced = a₂ − a₁
nth termaₙ = a + (n − 1)d
Last terml = a + (n − 1)d
Sum using a and dSₙ = n/2 [2a + (n − 1)d]
Sum using first and last termsSₙ = n/2(a + l)
Term from consecutive sumsaₙ = Sₙ − Sₙ₋₁
First n positive integersn(n + 1)/2
Arithmetic meanb = (a + c)/2

Which Formula Should I Use?

Question TypeBest Approach
Is this sequence an AP?Compare consecutive differences
Find the 25th termaₙ = a + (n − 1)d
Which term is equal to a given number?Set aₙ equal to that number and solve for n
Find how many terms are presentUse the last term as aₙ and solve for n
Find the total of the first n termsUse Sₙ
First and last terms are knownUse Sₙ = n/2(a + l)
Sₙ and Sₙ₋₁ are knownUse aₙ = Sₙ − Sₙ₋₁
Two different terms are knownCreate two nth-term equations
Term or Sum?

Before touching a formula, ask one question: Do I need ONE term or the TOTAL of several terms? This single decision prevents many AP mistakes.

Write What You Know First

Problem-Solving Habit

For most AP problems, begin by writing a = ?, d = ?, n = ?, aₙ = ? and, when required, Sₙ = ?. Filling in these values before doing algebra makes complicated word problems much easier.

Mixed Worked Examples

Worked Example: Find a Particular Term

Problem
Find the 18th term of the AP 7, 12, 17, 22, ... .

  1. 1.Here a = 7.
  2. 2.d = 12 − 7 = 5.
  3. 3.n = 18.
  4. 4.Use aₙ = a + (n − 1)d.
  5. 5.a₁₈ = 7 + 17(5).
  6. 6.a₁₈ = 7 + 85.
  7. 7.a₁₈ = 92.
  8. 8.Therefore, the 18th term is 92.
Worked Example: Find the Position

Problem
Which term of the AP 11, 17, 23, 29, ... is 161?

  1. 1.Here a = 11 and d = 6.
  2. 2.Let 161 be the nth term.
  3. 3.161 = 11 + (n − 1)(6).
  4. 4.150 = 6(n − 1).
  5. 5.25 = n − 1.
  6. 6.n = 26.
  7. 7.Therefore, 161 is the 26th term.
Worked Example: Find a Sum

Problem
Find the sum of the first 25 terms of the AP 4, 9, 14, 19, ... .

  1. 1.Here a = 4, d = 5 and n = 25.
  2. 2.Use Sₙ = n/2 [2a + (n − 1)d].
  3. 3.S₂₅ = 25/2 [8 + 24(5)].
  4. 4.S₂₅ = 25/2 [128].
  5. 5.S₂₅ = 25 × 64.
  6. 6.S₂₅ = 1600.
  7. 7.Therefore, the sum of the first 25 terms is 1600.
Worked Example: Find Missing a and d

Problem
The 4th term of an AP is 16 and the 9th term is 36. Find its first term and common difference.

  1. 1.For the 4th term, a + 3d = 16. ...(1)
  2. 2.For the 9th term, a + 8d = 36. ...(2)
  3. 3.Subtract equation (1) from equation (2).
  4. 4.5d = 20.
  5. 5.d = 4.
  6. 6.Substitute d = 4 into a + 3d = 16.
  7. 7.a + 12 = 16.
  8. 8.a = 4.
  9. 9.Therefore, the first term is 4 and common difference is 4.

Common Mistakes to Avoid

Common Mistakes

• Calculating d as earlier term − later term instead of later term − earlier term. • Using a + nd instead of a + (n − 1)d. • Confusing n with aₙ. • Confusing one term aₙ with the total Sₙ. • Losing the negative sign when d < 0. • Accepting a fractional, zero or negative value of n as a term position. • Forgetting the +1 when converting a position from the end. • Using a sum formula without first determining the correct number of terms. • Assuming every number pattern is an AP.

A Simple Strategy for AP Problems

StepWhat to Do
1Write the sequence if the problem is given in words.
2Identify a and calculate d.
3Decide whether the question asks for one term or a sum.
4Write the relevant formula before substituting values.
5Solve carefully and check whether the answer makes sense.

Chapter Practice

Part A — Foundations

Practice Problems

Foundation Practice
  1. Find the first term and common difference of the AP 8, 13, 18, 23, ... .
  2. Find the first term and common difference of the AP 31, 25, 19, 13, ... .
  3. Write the first four terms of an AP whose first term is −5 and common difference is 4.
  4. Check whether 3, 7, 11, 15, ... is an Arithmetic Progression.
  5. Check whether 2, 6, 18, 54, ... is an Arithmetic Progression.
  6. Write the next three terms of the AP 42, 35, 28, 21, ... .

Part B — nth-Term Practice

nth-Term Practice
  1. Find the 32nd term of the AP 14, 10, 6, 2, ... .
  2. Which term of the AP 6, 11, 16, 21, ... is 126?
  3. Check whether −127 is a term of the AP 17, 13, 9, 5, ... .
  4. Find the number of terms in the AP 9, 15, 21, ... , 207.
  5. The 12th term of an AP is 41 and the 18th term is 65. Find the 30th term.
  6. The 4th term of an AP is 13 and the 10th term is −5. Which term of the AP is zero?
  7. Find the 15th term from the end of the AP 4, 9, 14, ... , 259.
  8. How many three-digit positive integers are divisible by 9?
  9. How many multiples of 6 lie strictly between 20 and 350?
  10. For what value of n are the nth terms of the APs 15, 19, 23, ... and 6, 13, 20, ... equal?

Part C — Sum Practice

Sum Practice
  1. Find the sum of the first 18 terms of the AP 4, 10, 16, 22, ... .
  2. Find the sum of the first 16 terms of the AP −25, −20, −15, ... .
  3. Find the sum 48 + 44 + 40 + ... + 8.
  4. An AP has first term 6, last term 66 and common difference 4. Find the number of terms and their sum.
  5. The first term of an AP is 8, the last term is 68 and the sum is 456. Find the number of terms and common difference.
  6. Find the sum of the first 30 positive integers divisible by 7.
  7. Find the sum of the first 20 positive multiples of 9.
  8. Find the sum of all odd positive integers less than 80.
  9. The second and third terms of an AP are 17 and 22 respectively. Find the sum of its first 40 terms.
  10. If the sum of the first n terms of a sequence is Sₙ = 5n − n², find its first term, second term and nth term.

Part D — Application Problems

Application Practice
  1. A construction company charges a penalty of ₹300 for the first day of delay, ₹375 for the second day, ₹450 for the third day and so on, increasing the penalty by ₹75 each day. Find the total penalty for a delay of 24 days.
  2. A school has ₹1,080 to distribute as 8 merit prizes. Each prize is ₹20 less than the prize before it. Find the value of all eight prizes.
  3. A school plans a plantation drive. Each section of Class I plants 2 trees, each section of Class II plants 4 trees, each section of Class III plants 6 trees and so on up to Class X. If there are 4 sections in every class, find the total number of trees planted.
  4. A stack contains 210 wooden blocks. The bottom row contains 24 blocks, the next row contains 23 blocks, then 22 blocks and so on. Find the number of rows and the number of blocks in the top row.
  5. In a collection race, a basket is placed 4 m from the first marker. The remaining markers are placed 2 m apart in a straight line. There are 12 markers. A participant runs from the basket to each marker separately and returns the marker to the basket before collecting the next one. Find the total distance travelled.
  6. The lengths of the horizontal bars of a decorative frame decrease uniformly from 60 cm at the bottom to 36 cm at the top. If successive bars differ in length by 3 cm, find the number of bars and the total length of material required.

Part E — Challenge Practice

Quiz

Quick check

What is the common difference of 18, 14, 10, 6, ... ?

Quick check

Which formula gives the nth term of an AP?

Quick check

What does Sₙ represent?

Quick check

Which statement is correct for 9, 9, 9, 9, ... ?

Quick check

Which relation can be used to find one term from two consecutive partial sums?

Quick check

If the first term, last term and number of terms are known, which expression is most convenient for finding the sum?

Self-Check

Before moving on, make sure you can answer these without looking at the lesson: • What condition makes a sequence an AP? • How do you calculate the common difference? • Why does the nth-term formula contain n − 1? • What is the difference between aₙ and Sₙ? • When should you use Sₙ = n/2(a + l)? • Why must a value of n represent a positive whole-number position?

Key Takeaways

Key Takeaways

• An AP has a constant difference between consecutive terms. • Its general form is a, a + d, a + 2d, a + 3d, ... . • The nth term is aₙ = a + (n − 1)d. • The sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d]. • If the last term is known, Sₙ = n/2(a + l) may be more convenient. • A term can be recovered from sums using aₙ = Sₙ − Sₙ₋₁. • Most word problems become easier once a, d, n and the required quantity are identified clearly. • Always decide whether the question asks for one term or the total of several terms before selecting a formula.

Previous · Lesson 3

Sum of First n Terms of an AP

Next

End of chapter