Arithmetic Progressions notes

5

Arithmetic Progressions

Detailed Theory • Easy English • Board Exam Focus

Join WhatsApp Join Telegram Visit Website

Chapter Roadmap

1. Sequences and Patterns 2. Arithmetic Progression 3. Common Difference 4. nth Term of an AP 5. Sum of First n Terms 6. Finding Missing Terms 7. AP Word Problems 8. Board-Style Questions 9. Board Exam Strategy 10. Final 96% Target Revision

✍️ For More Notes Visit UniversityScope.com ✍️

1. Sequences and Patterns

What is a Sequence?

A sequence is a list of numbers written in a definite order according to a rule.

\[\boxed{2,\;5,\;8,\;11,\;14,\ldots}\]

Each number in a sequence is called a term. We commonly write the terms as \(a_1,a_2,a_3,\ldots\).

Solved Example 1 — Identify the terms

Question: Write the first four terms of the sequence \(3,7,11,15,\ldots\).

1. The first term is \(3\).
2. The second term is \(7\).
3. The third term is \(11\).
4. The fourth term is \(15\).
5. Answer: The first four terms are \(\boxed{3,7,11,15}\).

Solved Example 2 — Find the pattern

Question: Find the next two terms of \(4,9,14,19,\ldots\).

1. Compare consecutive terms: \(9-4=5\).
2. Again, \(14-9=5\).
3. Again, \(19-14=5\).
4. So \(5\) is added each time.
5. Next term \(=19+5=24\).
6. Next term \(=24+5=29\).
7. Answer: \(\boxed{24,29}\).

Solved Example 3 — Find a term

Question: In the sequence \(10,20,30,40,\ldots\), what is the fifth term?

1. The terms are \(10,20,30,40,\ldots\).
2. Each term increases by \(10\).
3. After \(40\), the next term is \(40+10\).
4. So the fifth term is \(50\).
5. Answer: \(\boxed{50}\).

2. Arithmetic Progression

Meaning of Arithmetic Progression

An Arithmetic Progression (AP) is a sequence in which the difference between any two consecutive terms is always the same.

\[\boxed{a,\;a+d,\;a+2d,\;a+3d,\ldots}\]

Here \(a\) is the first term and \(d\) is the common difference.

Solved Example 1 — Check an AP

Question: Is \(5,8,11,14,17\) an AP?

1. Subtract the first term from the second: \(8-5=3\).
2. Subtract the second from the third: \(11-8=3\).
3. Subtract the third from the fourth: \(14-11=3\).
4. Subtract the fourth from the fifth: \(17-14=3\).
5. The difference is constant.
6. Answer: Yes, it is an AP with \(d=3\).

Solved Example 2 — Not an AP

Question: Check whether \(2,4,8,16\) is an AP.

1. \(4-2=2\).
2. \(8-4=4\).
3. The differences are already different.
4. Therefore, there is no common difference.
5. Answer: \(\boxed{\text{Not an AP}}\).

Solved Example 3 — Identify \(a\) and \(d\)

Question: For \(12,17,22,27,\ldots\), find the first term and common difference.

1. The first term is the first number.
2. So \(a=12\).
3. Find the difference: \(17-12=5\).
4. Check: \(22-17=5\).
5. Therefore \(d=5\).
6. Answer: \(\boxed{a=12,\ d=5}\).

3. Common Difference

Formula for Common Difference

\[\boxed{d=a_2-a_1}\]

For an AP, the same value is obtained from every pair of consecutive terms.

Solved Example 1 — Positive difference

Question: Find \(d\) for \(7,12,17,22,\ldots\).

1. Take the second term: \(12\).
2. Subtract the first term: \(7\).
3. \(d=12-7=5\).
4. Answer: \(\boxed{d=5}\).

Solved Example 2 — Negative difference

Question: Find \(d\) for \(20,15,10,5,\ldots\).

1. Take the second term: \(15\).
2. Subtract the first term: \(20\).
3. \(d=15-20=-5\).
4. Answer: \(\boxed{d=-5}\).

Solved Example 3 — Fractional difference

Question: Find the common difference of \(\frac12,\frac32,\frac52,\frac72,\ldots\).

1. Use \(d=a_2-a_1\).
2. \(d=\frac32-\frac12\).
3. \(d=\frac{3-1}{2}=1\).
4. Answer: \(\boxed{d=1}\).

4. nth Term of an Arithmetic Progression

Important Formula

If the first term is \(a\), common difference is \(d\), and the number of terms is \(n\), then the \(n\)th term is

\[\boxed{a_n=a+(n-1)d}\]

Solved Example 1 — Find the 10th term

Question: Find the 10th term of \(3,7,11,15,\ldots\).

1. First term \(a=3\).
2. Common difference \(d=7-3=4\).
3. We need \(n=10\).
4. Use \(a_n=a+(n-1)d\).
5. \(a_{10}=3+(10-1)(4)\).
6. \(=3+36=39\).
7. Answer: \(\boxed{39}\).

Solved Example 2 — Find a large term

Question: Find the 25th term of \(8,11,14,17,\ldots\).

1. First term \(a=8\).
2. Common difference \(d=11-8=3\).
3. Here \(n=25\).
4. \(a_{25}=8+(25-1)(3)\).
5. \(=8+24\times3\).
6. \(=8+72=80\).
7. Answer: \(\boxed{80}\).

Solved Example 3 — Negative common difference

Question: Find the 12th term of \(30,27,24,21,\ldots\).

1. First term \(a=30\).
2. Common difference \(d=27-30=-3\).
3. Here \(n=12\).
4. \(a_{12}=30+(12-1)(-3)\).
5. \(=30-33\).
6. \(=-3\).
7. Answer: \(\boxed{-3}\).

5. Sum of First \(n\) Terms of an AP

Sum Formula

The sum of the first \(n\) terms of an AP is

\[\boxed{S_n=\frac n2\left[2a+(n-1)d\right]}\]

If the last term \(l\) is known, we can also use

\[\boxed{S_n=\frac n2(a+l)}\]

Solved Example 1 — Direct sum

Question: Find the sum of the first 10 terms of \(2,5,8,\ldots\).

1. First term \(a=2\).
2. Common difference \(d=3\).
3. Number of terms \(n=10\).
4. Use \(S_n=\frac n2[2a+(n-1)d]\).
5. \(S_{10}=\frac{10}{2}[2(2)+9(3)]\).
6. \(=5[4+27]=5(31)\).
7. \(=155\).
8. Answer: \(\boxed{155}\).

Solved Example 2 — Using last term

Question: Find the sum of \(5,10,15,\ldots,50\).

1. First term \(a=5\).
2. Common difference \(d=5\).
3. Last term \(l=50\).
4. Find the number of terms using \(a_n=a+(n-1)d\).
5. \(50=5+(n-1)5\).
6. \(45=5(n-1)\), so \(n-1=9\).
7. Therefore \(n=10\).
8. Use \(S_n=\frac n2(a+l)\).
9. \(S_{10}=\frac{10}{2}(5+50)=5(55)=275\).
10. Answer: \(\boxed{275}\).

Solved Example 3 — Sum of first terms

Question: Find the sum of the first 20 natural numbers.

1. The AP is \(1,2,3,\ldots,20\).
2. So \(a=1,\ d=1,\ n=20\).
3. Use \(S_n=\frac n2[2a+(n-1)d]\).
4. \(S_{20}=\frac{20}{2}[2(1)+19(1)]\).
5. \(=10(21)\).
6. \(=210\).
7. Answer: \(\boxed{210}\).

6. Finding Missing Terms in an AP

Basic Idea

If terms of an AP are missing, use the fact that the difference between consecutive terms is constant.

Solved Example 1 — One missing term

Question: Find \(x\) if \(5,x,17\) are consecutive terms of an AP.

1. For consecutive terms, the differences are equal.
2. So \(x-5=17-x\).
3. Add \(x\) to both sides: \(2x-5=17\).
4. Add \(5\): \(2x=22\).
5. Divide by \(2\): \(x=11\).
6. Answer: \(\boxed{x=11}\).

Solved Example 2 — Three missing terms

Question: Find \(x,y,z\) if \(4,x,y,z,20\) are consecutive terms of an AP.

1. There are four equal gaps from \(4\) to \(20\).
2. Total increase \(=20-4=16\).
3. Each gap \(=16/4=4\).
4. So \(x=4+4=8\).
5. \(y=8+4=12\).
6. \(z=12+4=16\).
7. Answer: \(\boxed{x=8,\ y=12,\ z=16}\).

Solved Example 3 — Find a term from conditions

Question: If \(7,x,19\) are consecutive terms of an AP, find \(x\).

1. Common differences must be equal.
2. \(x-7=19-x\).
3. \(2x=26\).
4. \(x=13\).
5. Check: \(13-7=6\) and \(19-13=6\).
6. Answer: \(\boxed{13}\).

7. AP Word Problems

Basic Approach

Read the situation carefully, choose the first term and common difference, identify the required number of terms, and then apply the correct AP formula.

Solved Example 1 — Seating arrangement

Question: The number of seats in successive rows of a hall are \(20,24,28,\ldots\). How many seats are in the 15th row?

1. First term \(a=20\).
2. Common difference \(d=24-20=4\).
3. We need the 15th term, so \(n=15\).
4. \(a_{15}=20+(15-1)4\).
5. \(=20+56=76\).
6. Answer: \(\boxed{76\text{ seats}}\).

Solved Example 2 — Savings

Question: A student saves ₹50 in the first week and increases the saving by ₹10 every week. How much is saved in the 12th week?

1. First week's saving \(a=50\).
2. Increase each week \(d=10\).
3. Required week \(n=12\).
4. \(a_{12}=50+(12-1)(10)\).
5. \(=50+110=160\).
6. Answer: \(\boxed{₹160}\).

Solved Example 3 — Total saving

Question: A student saves ₹100 in the first month and ₹20 more every month. Find the total saving in the first 12 months.

1. First term \(a=100\).
2. Common difference \(d=20\).
3. Number of terms \(n=12\).
4. Use \(S_n=\frac n2[2a+(n-1)d]\).
5. \(S_{12}=6[200+11(20)]\).
6. \(=6(200+220)=6(420)\).
7. \(=2520\).
8. Answer: \(\boxed{₹2520}\).

8. Fully Solved Board-Style Questions

Question 1. Find the 20th term of \(6,10,14,\ldots\).

Solution:

\(a=6,\ d=4,\ n=20\).

\[ a_{20}=6+(20-1)(4)=6+76=82 \]

Answer: \(\boxed{82}\).

Question 2. Find the sum of the first 15 terms of \(4,7,10,\ldots\).

Solution:

\(a=4,\ d=3,\ n=15\).

\[ S_{15}=\frac{15}{2}[2(4)+14(3)] \]

\[ =\frac{15}{2}(8+42)=\frac{15}{2}(50)=375 \]

Answer: \(\boxed{375}\).

Question 3. Which term of the AP \(5,9,13,17,\ldots\) is \(81\)?

Solution:

Here \(a=5,\ d=4\). Let \(a_n=81\).

\[ 81=5+(n-1)4 \]

\[ 76=4(n-1) \]

\[ 19=n-1\Rightarrow n=20 \]

Answer: \(81\) is the \(\boxed{20\text{th}}\) term.

Question 4. Find three consecutive terms of an AP whose sum is \(27\).

Solution:

Let the three consecutive terms be \(a-d,\ a,\ a+d\).

\[ (a-d)+a+(a+d)=27 \]

\[ 3a=27 \]

\[ a=9 \]

Therefore the terms are \(9-d,9,9+d\). A unique set cannot be found without one more condition.

Answer: The middle term is \(\boxed{9}\); one additional condition is required for the exact three terms.

Question 5. Find the sum of all multiples of \(5\) from \(5\) to \(100\).

Solution:

The AP is \(5,10,15,\ldots,100\).

Here \(a=5,\ d=5,\ l=100\).

\[ 100=5+(n-1)5 \]

\[ 95=5(n-1)\Rightarrow n=20 \]

\[ S_{20}=\frac{20}{2}(5+100)=10(105)=1050 \]

Answer: \(\boxed{1050}\).

9. How to Write Arithmetic Progression Answers in the Board Exam

1. Write the given AP clearly.
2. Identify the first term \(a\).
3. Calculate the common difference \(d\).
4. Write the required term number \(n\).
5. Choose the correct formula.
6. Substitute values carefully.
7. Show all important calculation steps.
8. Write the final answer in a box with units when required.

Common Mistakes to Avoid

• Using \(n d\) instead of \((n-1)d\) in the nth-term formula.
• Taking the wrong sign of the common difference.
• Confusing \(a_n\) with \(S_n\).
• Forgetting to find \(n\) when the last term is given.
• Using the sum formula when the question asks for one term.
• Ignoring units in word problems.

10. Final 96% Target Revision

Must-Remember Formulas

\[\boxed{a,\ a+d,\ a+2d,\ldots}\]

\[\boxed{a_n=a+(n-1)d}\]

\[\boxed{S_n=\frac n2[2a+(n-1)d]}\]

\[\boxed{S_n=\frac n2(a+l)}\]

Quick facts: \(a\) = first term, \(d\) = common difference, \(n\) = number of terms, \(a_n\) = nth term, \(S_n\) = sum of first \(n\) terms.

One-Minute Checklist

• ✓ Know the meaning of an arithmetic progression.
• ✓ Find the common difference correctly.
• ✓ Use \(a_n=a+(n-1)d\) for the nth term.
• ✓ Use the correct sum formula.
• ✓ Find the number of terms when needed.
• ✓ Handle negative and fractional common differences.
• ✓ Solve AP word problems step by step.
• ✓ Show complete working in board answers.

✍️ For More Notes Visit UniversityScope.com ✍️

Join WhatsApp Join Telegram Visit Website

Class 10 Maths • Chapter 5: Arithmetic Progressions

Designed as easy-to-read handwritten-style board exam notes.