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Lesson 3 of 4

Arithmetic Progressions · Lesson 3 of 4

Sum of First n Terms of an AP

“If you're still adding terms one by one, Carl Friedrich Gauss is shaking his head in mathematical disappointment.”

Learning Objectives

• Understand why a shortcut is needed to find the sum of many terms of an AP. • Derive and use the formula for the sum of the first n terms of an AP. • Use the alternative sum formula when the first and last terms are known. • Find unknown quantities such as n, a or d using the sum formula. • Apply AP sums to number patterns and real-life problems.

In the previous lesson, we learned how to jump directly to any term of an Arithmetic Progression. But sometimes we do not want just one term. We want the total of many terms.

Suppose a student saves ₹100 in the first month, ₹150 in the second month, ₹200 in the third month and continues increasing the amount saved by ₹50 every month.

Monthly SavingsLaTeX
The monthly savings form an Arithmetic Progression with a = 100 and d = 50.
A New Kind of Question

Finding the amount saved in the 20th month is an nth-term problem. But finding the total amount saved during the first 20 months requires us to add all 20 terms. Can we do this without adding them one by one?

Definition
Sum of First n Terms

The sum of the first n terms of an Arithmetic Progression is represented by Sₙ.

Sum NotationLaTeX
For example, S₁₀ means the sum of the first 10 terms and S₂₀ means the sum of the first 20 terms.

Gauss and a Clever Shortcut

A famous story tells us that the mathematician Carl Friedrich Gauss was once asked, as a child, to add all the whole numbers from 1 to 100. Instead of adding them one by one, he noticed a beautiful pattern.

Write the Numbers NormallyLaTeX
Write Them in ReverseLaTeX

Now add the two rows vertically. The first pair gives 1 + 100 = 101. The second gives 2 + 99 = 101. Every pair gives exactly 101.

Pairing the TermsLaTeX
There are 100 such pairs, each equal to 101.
Gauss's ResultLaTeX
The Key Idea

Gauss did not really make addition disappear. He reorganised the numbers so that a repeating pattern became visible. We can use the same idea for any Arithmetic Progression.

Deriving the Sum Formula

Consider an AP whose first term is a, common difference is d and which contains n terms.

The APLaTeX
Write the SumLaTeX
Reverse the SumLaTeX
Now write exactly the same terms in reverse order.

Now add the two equations term by term. Each pair gives the same result.

First PairLaTeX
Second PairLaTeX
What Happens When We Pair?

Every pair has the same value: 2a + (n − 1)d. Since there are n terms, this same value appears n times.

DerivationLaTeX
Sum of First n Terms of an APLaTeX
Use this form when a, d and n are known.

Understanding the Formula

SymbolMeaning
SₙSum of the first n terms
aFirst term
dCommon difference
nNumber of terms
A Good Habit

Before using the sum formula, write down a = ?, d = ?, n = ? and Sₙ = ?. This prevents most substitution mistakes.

Finding the Sum Directly

Worked Example: Find the Sum of 10 Terms

Problem
Find the sum of the first 10 terms 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.The number of terms is n = 10.
  4. 4.Use Sₙ = n/2 [2a + (n − 1)d].
  5. 5.S₁₀ = 10/2 [2(2) + (10 − 1)(5)].
  6. 6.S₁₀ = 5[4 + 45].
  7. 7.S₁₀ = 5 × 49.
  8. 8.S₁₀ = 245.
  9. 9.Therefore, the sum of the first 10 terms is 245.
CalculationLaTeX

The Sum Can Also Be Negative

An AP may decrease and eventually contain negative numbers. In that case, the total sum can also become negative.

Worked Example: Sum with a Negative Common Difference

Problem
Find the sum of the first 22 terms of the AP 8, 3, −2, ... .

  1. 1.The first term is a = 8.
  2. 2.The common difference is d = 3 − 8 = −5.
  3. 3.The number of terms is n = 22.
  4. 4.Use Sₙ = n/2 [2a + (n − 1)d].
  5. 5.S₂₂ = 22/2 [2(8) + 21(−5)].
  6. 6.S₂₂ = 11[16 − 105].
  7. 7.S₂₂ = 11(−89).
  8. 8.S₂₂ = −979.
  9. 9.Therefore, the sum of the first 22 terms is −979.
Do Not Fear a Negative Sum

A negative answer does not automatically mean the calculation is wrong. If later negative terms outweigh the earlier positive terms, the total can genuinely be negative.

When the Last Term Is Known

Sometimes the common difference is not needed because the first term, last term and number of terms are already known.

The Last TermLaTeX

Since a + (n − 1)d is the last term l, we can replace that part of the sum formula.

Alternative Sum FormulaLaTeX
This form is especially useful when a, l and n are given.
FormulaBest Used When
Sₙ = n/2 [2a + (n − 1)d]When a, d and n are known
Sₙ = n/2 (a + l)When a, l and n are known
Worked Example: First and Last Terms Are Known

Problem
Find the sum of the AP 5, 10, 15, ... , 100.

  1. 1.Here a = 5.
  2. 2.The common difference is d = 5.
  3. 3.The last term is l = 100.
  4. 4.First find the number of terms.
  5. 5.100 = 5 + (n − 1)(5).
  6. 6.95 = 5(n − 1).
  7. 7.19 = n − 1.
  8. 8.n = 20.
  9. 9.Now use Sₙ = n/2(a + l).
  10. 10.S₂₀ = 20/2(5 + 100).
  11. 11.S₂₀ = 10 × 105.
  12. 12.S₂₀ = 1050.
  13. 13.Therefore, the sum is 1050.
Which Formula Should I Use?

Both sum formulas are equivalent. Do not memorise them as unrelated formulas. Choose the one that uses the information already given in the question.

Returning to the Savings Problem

Suppose ₹100 is saved in the first month and the amount increases by ₹50 every month. How much is saved in total during the first 21 months?

Worked Example: Total Savings

Problem
Find the total of the first 21 terms of 100, 150, 200, 250, ... .

  1. 1.Here a = 100.
  2. 2.The common difference is d = 50.
  3. 3.The number of terms is n = 21.
  4. 4.Use Sₙ = n/2 [2a + (n − 1)d].
  5. 5.S₂₁ = 21/2 [2(100) + 20(50)].
  6. 6.S₂₁ = 21/2 [200 + 1000].
  7. 7.S₂₁ = 21/2 × 1200.
  8. 8.S₂₁ = 12,600.
  9. 9.Therefore, the total amount saved is ₹12,600.

Using the Sum Formula Backwards

The formula contains four quantities: Sₙ, a, d and n. If any three are known, we can usually find the fourth.

Question TypeKnownFind
Find the suma, d, nSₙ
Find the first termSₙ, d, na
Find the common differenceSₙ, a, nd
Find the number of termsSₙ, a, dn
Worked Example: Finding d from the Sum

Problem
The sum of the first 14 terms of an AP is 1050 and its first term is 10. Find the common difference.

  1. 1.Here S₁₄ = 1050, a = 10 and n = 14.
  2. 2.Use Sₙ = n/2 [2a + (n − 1)d].
  3. 3.1050 = 14/2 [20 + 13d].
  4. 4.1050 = 7(20 + 13d).
  5. 5.150 = 20 + 13d.
  6. 6.130 = 13d.
  7. 7.d = 10.
  8. 8.Therefore, the common difference is 10.
Solving for dLaTeX
Using the Result to Find Another Term

Problem
Using a = 10 and d = 10 from the previous example, find the 20th term.

  1. 1.Use the nth-term formula aₙ = a + (n − 1)d.
  2. 2.a₂₀ = 10 + (20 − 1)(10).
  3. 3.a₂₀ = 10 + 190.
  4. 4.a₂₀ = 200.
  5. 5.Therefore, the 20th term is 200.

Finding n When the Sum Is Known

When Sₙ is given and n is unknown, substituting into the sum formula may produce a quadratic equation. Solving that equation gives the possible number of terms.

Worked Example: How Many Terms Give a Sum of 78?

Problem
How many terms of the AP 24, 21, 18, ... must be taken so that their sum is 78?

  1. 1.Here a = 24.
  2. 2.The common difference is d = −3.
  3. 3.The required sum is Sₙ = 78.
  4. 4.Use Sₙ = n/2 [2a + (n − 1)d].
  5. 5.78 = n/2 [48 − 3(n − 1)].
  6. 6.78 = n/2 [51 − 3n].
  7. 7.156 = 51n − 3n².
  8. 8.3n² − 51n + 156 = 0.
  9. 9.Divide by 3: n² − 17n + 52 = 0.
  10. 10.Factorise: (n − 4)(n − 13) = 0.
  11. 11.Therefore, n = 4 or n = 13.
  12. 12.Both values are valid.
Why Can There Be Two Answers?

It may seem strange that two different numbers of terms can give the same sum. This can happen when an AP starts with positive terms and later contains negative terms. Some later terms may cancel the earlier total.

Check Every Value of n

When solving for n, keep only values that make sense as a number of terms. n must be a positive integer. However, unlike many simple problems, more than one positive integer answer can sometimes be valid.

Finding a Term from Two Sums

There is another useful relationship between terms and sums. The sum Sₙ contains the first n terms, while Sₙ₋₁ contains only the first n − 1 terms.

Compare the Two SumsLaTeX
Term from SumsLaTeX
Subtracting removes all the earlier terms and leaves only the nth term.
Worked Example: Finding a Term from Sums

Problem
If S₁₀ = 250 and S₉ = 207, find the 10th term.

  1. 1.Use aₙ = Sₙ − Sₙ₋₁.
  2. 2.a₁₀ = S₁₀ − S₉.
  3. 3.a₁₀ = 250 − 207.
  4. 4.a₁₀ = 43.
  5. 5.Therefore, the 10th term is 43.

Sum of the First n Positive Integers

The positive integers 1, 2, 3, 4, ... themselves form an AP with first term 1 and common difference 1.

Sum of First n Positive IntegersLaTeX
This is an important special case of the AP sum formula.
Worked Example: Sum of First 1000 Positive Integers

Problem
Find 1 + 2 + 3 + ... + 1000.

  1. 1.Here n = 1000.
  2. 2.Use n(n + 1)/2.
  3. 3.Sum = 1000(1001)/2.
  4. 4.Sum = 500 × 1001.
  5. 5.Sum = 500500.
  6. 6.Therefore, the sum of the first 1000 positive integers is 500500.

Sums of Multiples

Multiples of a fixed number form an AP. This means the sum formula can be used to add many multiples quickly.

Worked Example: Sum of the First 15 Multiples of 8

Problem
Find the sum of the first 15 positive multiples of 8.

  1. 1.The sequence is 8, 16, 24, ... .
  2. 2.Here a = 8 and d = 8.
  3. 3.There are n = 15 terms.
  4. 4.Use Sₙ = n/2 [2a + (n − 1)d].
  5. 5.S₁₅ = 15/2 [16 + 14(8)].
  6. 6.S₁₅ = 15/2 [16 + 112].
  7. 7.S₁₅ = 15/2 × 128.
  8. 8.S₁₅ = 960.
  9. 9.Therefore, the sum is 960.

When the nth Term Is Given

Sometimes an AP is not written as a list. Instead, its nth term is given by a formula. In that case, first generate a few terms and identify a and d.

Worked Example: Sum from an nth-Term Rule

Problem
The nth term of a sequence is aₙ = 3 + 2n. Find the sum of the first 24 terms.

  1. 1.Find the first few terms.
  2. 2.a₁ = 3 + 2(1) = 5.
  3. 3.a₂ = 3 + 2(2) = 7.
  4. 4.a₃ = 3 + 2(3) = 9.
  5. 5.So the sequence is 5, 7, 9, 11, ... .
  6. 6.The common difference is d = 2.
  7. 7.Here a = 5 and n = 24.
  8. 8.Use Sₙ = n/2 [2a + (n − 1)d].
  9. 9.S₂₄ = 24/2 [10 + 23(2)].
  10. 10.S₂₄ = 12(56).
  11. 11.S₂₄ = 672.
  12. 12.Therefore, the sum of the first 24 terms is 672.

Using AP Sums in Real Life

Whenever a quantity changes by a fixed amount and we need the total over several stages, an AP sum may be hiding inside the problem.

SituationPossible AP Sum Question
Money saved increases by a fixed amountAdd all savings
Production increases by the same amount every yearFind total production
A daily penalty increases by a fixed amountFind total penalty
Prize money decreases by a fixed amountFind total prize money
Rows contain increasing or decreasing quantitiesFind total items
Worked Example: Production Over Several Years

Problem
A factory produces 600 units in the 3rd year and 700 units in the 7th year. Production increases by a fixed amount every year. Find the total production during the first 7 years.

  1. 1.Let the first-year production be a and the yearly increase be d.
  2. 2.The 3rd term is 600, so a + 2d = 600.
  3. 3.The 7th term is 700, so a + 6d = 700.
  4. 4.Subtract the equations: 4d = 100.
  5. 5.Therefore, d = 25.
  6. 6.Substitute into a + 2d = 600.
  7. 7.a + 50 = 600, so a = 550.
  8. 8.Now use S₇ = 7/2 [2a + (7 − 1)d].
  9. 9.S₇ = 7/2 [1100 + 150].
  10. 10.S₇ = 7/2 × 1250.
  11. 11.S₇ = 4375.
  12. 12.Therefore, the factory produces a total of 4375 units during the first 7 years.

Term or Sum? Choose Carefully

QuestionWhat Is Asked?Use
Find the 20th termOne valueaₙ = a + (n − 1)d
Find the total of first 20 termsTotalSₙ formula
Salary in the 10th yearOne year's salarynth-term formula
Total salary increase over several stagesTotal of several valuesSum formula
Number of plants in the last rowOne termnth-term formula
Total plants in all rowsTotalSum formula
Most Important Distinction

Do not confuse aₙ with Sₙ. aₙ means one particular term. Sₙ means the total of the first n terms.

Common Mistakes

• Using the nth-term formula when the question asks for a total. • Confusing aₙ with Sₙ. • Forgetting the factor n/2 in the sum formula. • Using n instead of n − 1 inside the bracket. • Losing the negative sign when d is negative. • Using the formula Sₙ = n/2(a + l) before finding the correct number of terms. • Accepting an invalid value of n without checking whether it is a positive integer. • Assuming that a negative sum must be incorrect.

Quiz

Quick check

Which formula gives the sum of the first n terms of an AP?

Quick check

When is Sₙ = n/2(a + l) especially useful?

Quick check

What is the sum of the first 10 positive integers?

Quick check

How can the nth term be found using consecutive partial sums?

Quick check

What is the correct difference between aₙ and Sₙ?

Quick check

What is the sum of the first n positive integers?

Practice Problems

Practice Questions
  1. Find the sum of the first 10 terms of the AP 2, 7, 12, 17, ... .
  2. Find the sum of the first 12 terms of the AP −37, −33, −29, ... .
  3. Find the sum of the first 20 terms of the AP 5, 9, 13, 17, ... .
  4. Find the sum 34 + 32 + 30 + ... + 10.
  5. An AP has first term 5, common difference 3 and last term 50. Find the number of terms and their sum.
  6. The sum of the first 10 terms of an AP is 125 and its third term is 15. Find the common difference.
  7. Find the sum of the first 40 positive integers divisible by 6.
  8. Find the sum of the first 15 multiples of 8.
  9. Find the sum of all odd numbers between 0 and 50.
  10. A contractor pays a penalty of ₹200 on the first day of delay, ₹250 on the second day, ₹300 on the third day and so on. Find the total penalty for a delay of 30 days.

Key Takeaways

Key Takeaways

• Sₙ represents the sum of the first n terms of an Arithmetic Progression. • The main sum formula is Sₙ = n/2 [2a + (n − 1)d]. • If the last term l is known, we can use Sₙ = n/2(a + l). • Both sum formulas come from pairing the first and last terms of the AP. • The nth term can be recovered from consecutive sums using aₙ = Sₙ − Sₙ₋₁. • The sum formula can be used backwards to find a, d or n when enough information is given. • The sum of the first n positive integers is n(n + 1)/2. • Always distinguish between aₙ, which represents one term, and Sₙ, which represents a total.

Previous · Lesson 2

nth Term of an AP

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Chapter Summary and Practice