Arithmetic Progressions (Class 10): Notes और MCQ Practice

आसान भाषा में:

Arithmetic Progression (AP) संख्याओं की ऐसी list है, जहाँ हर अगला term पिछले term में एक निश्चित संख्या (fixed number) जोड़कर मिलता है। इस निश्चित संख्या को Common Difference (d) कहते हैं। जैसे: 2, 5, 8, 11... यहाँ हर step पर +3 जुड़ रहा है!

Important Formulas & Definitions:

• First Term (a) और Common Difference (d): d = a₂ − a₁ = aₙ − aₙ₋₁

(d positive, negative या zero कुछ भी हो सकता है।)

• n-th Term (General Term):

aₙ = a + (n − 1)d

• Sum of First n Terms (Sₙ):

Sₙ = (n ÷ 2) × [2a + (n − 1)d]

या Sₙ = (n ÷ 2) × (a + l) (जहाँ l = last term = aₙ)

• nth term from sum: aₙ = Sₙ − Sₙ₋₁

2 Memory Tricks:

1. (n − 1) Rule: याद रखें कि first term (a) में कोई d नहीं होता, इसलिए nth term तक पहुँचने के लिए हमेशा (n − 1) बार ही d जुड़ेगा।
2. Sum = Average × Count: Sₙ का दूसरा formula असल में first term और last term का average निकाल कर उसे n से multiply करना है: Sₙ = [(a + l) ÷ 2] × n.

2 Common Board-Exam Mistakes:

1. 'n' और 'aₙ' में Confusion: n हमेशा terms की संख्या है (जो हमेशा positive integer होगा: 1, 2, 3...), जबकि aₙ उस position की value है। अगर n fraction या negative आए, तो वह term AP में exist नहीं करता।
2. Common Difference (d) का Sign: d = a₂ − a₁ होता है। घटती हुई AP (जैसे: 10, 7, 4...) में d = 7 − 10 = −3 होगा। जल्दबाजी में sign छोड़ देने से पूरा answer गलत हो जाता है।

Arithmetic Progressions (AP) - Class 10 Study Notes

1. Chapter in Short

An Arithmetic Progression (AP) is a special list of numbers where each term is obtained by adding a fixed number to the preceding term (except the first term).

Instead of adding numbers one by one, this chapter teaches us how to:

• Predict any term in a sequence (like finding the 100th term directly).
• Find the sum of many terms quickly using simple formulas.

2. Key Concepts Explained Simply

What is an Arithmetic Progression?

Imagine climbing a ladder where each step is 20 cm above the previous one. If the first step is at 20 cm, the next steps are at 40 cm, 60 cm, 80 cm, and so on. Because the height increases by a fixed amount every time, this forms an Arithmetic Progression.

Everyday Example:

• Your piggy bank has 100 rupees. You add 50 rupees every week:
• Week 1: Rs 100
• Week 2: Rs 150
• Week 3: Rs 200
• Week 4: Rs 250
• The difference between consecutive weeks is always 50. This is an AP!

First Term (a) and Common Difference (d)

• First Term (a): The very first number in the sequence.
• Common Difference (d): The fixed number added to each term to get the next term.
• d = Second term − First term = a₂ − a₁
• Important: d can be positive (increasing AP), negative (decreasing AP), or zero (constant AP).

Why the n-th Term Formula Works

• 1st term = a
• 2nd term = a + d
• 3rd term = a + 2d
• 4th term = a + 3d

Notice that the number multiplying d is always 1 less than the term number. Therefore, to reach the n-th term, you add d exactly (n − 1) times.

Sum of First n Terms (Sₙ)

A famous mathematician named Carl Friedrich Gauss discovered that pairing numbers from both ends gives equal sums. Using this concept, the formula for the sum of n terms allows us to add hundreds of numbers in just a few seconds.

3. Important Definitions, Formulas, and Facts

• Standard Form of an AP:

a, a + d, a + 2d, a + 3d, ...

• Common Difference (d):

d = aₖ − aₖ₋₁ (Difference between any term and its preceding term)

• n-th Term of an AP (General Term):

aₙ = a + (n − 1)d

(where a = first term, d = common difference, n = number of terms)

• n-th Term from the End of a Finite AP:

Term from end = l − (n − 1)d

(where l is the last term)

• Sum of First n Terms (Sₙ):
• Formula 1 (when last term is unknown):

Sₙ = (n ÷ 2) × [2a + (n − 1)d]

• Formula 2 (when first term 'a' and last term 'l' are known):

Sₙ = (n ÷ 2) × (a + l)

• Sum of First n Positive Integers (1 + 2 + 3 + ... + n):

Sₙ = [n × (n + 1)] ÷ 2

• Finding aₙ using Sums:

aₙ = Sₙ − Sₙ₋₁

• Arithmetic Mean:

If a, b, c are in AP, then b = (a + c) ÷ 2, where b is called the arithmetic mean of a and c.

4. Solved Examples (Board-Exam Style)

Example 1: Finding a specific term

Question: Find the 30th term of the AP: 10, 7, 4, ...

Solution:

1. Identify the given values:

• First term, a = 10
• Common difference, d = 7 − 10 = −3
• Term number, n = 30

1. Use the n-th term formula:

aₙ = a + (n − 1)d

a₃₀ = 10 + (30 − 1) × (−3)

a₃₀ = 10 + 29 × (−3)

a₃₀ = 10 − 87 = −77

Answer: The 30th term is −77.

Example 2: Finding the number of terms (n)

Question: Which term of the AP: 3, 8, 13, 18, ... is 78?

Solution:

1. Identify given values:

• First term, a = 3
• Common difference, d = 8 − 3 = 5
• Let the n-th term be 78, so aₙ = 78

1. Substitute into the formula:

aₙ = a + (n − 1)d

78 = 3 + (n − 1) × 5

78 − 3 = (n − 1) × 5

75 = (n − 1) × 5

n − 1 = 75 ÷ 5 = 15

n = 15 + 1 = 16

Answer: The 16th term is 78.

Example 3: Finding the sum of terms

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

Solution:

1. Identify given values:

• a = 8
• d = 3 − 8 = −5
• n = 22

1. Use the sum formula:

Sₙ = (n ÷ 2) × [2a + (n − 1)d]

S₂₂ = (22 ÷ 2) × [2 × 8 + (22 − 1) × (−5)]

S₂₂ = 11 × [16 + 21 × (−5)]

S₂₂ = 11 × [16 − 105]

S₂₂ = 11 × (−89) = −979

Answer: The sum of the first 22 terms is −979.

Example 4: Finding n when sum is given

Question: How many terms of the AP: 9, 17, 25, ... must be taken to give a sum of 636?

Solution:

1. Identify given values:

• a = 9, d = 17 − 9 = 8, Sₙ = 636

1. Use the sum formula:

Sₙ = (n ÷ 2) × [2a + (n − 1)d]

636 = (n ÷ 2) × [2 × 9 + (n − 1) × 8]

636 = (n ÷ 2) × [18 + 8n − 8]

636 = (n ÷ 2) × [10 + 8n]

Divide inside terms by 2:

636 = n × (5 + 4n)

4n² + 5n − 636 = 0

1. Factor the quadratic equation:

4n² + 53n − 48n − 636 = 0

n(4n + 53) − 12(4n + 53) = 0

(n − 12)(4n + 53) = 0

• Either n = 12 or n = −53 ÷ 4

1. Since the number of terms n must be a positive integer, discard n = −53 ÷ 4.

Answer: 12 terms must be taken.

5. Common Mistakes and Quick Revision Points

Common Mistakes to Avoid:

1. Sign error in common difference (d):

• Always compute d = a₂ − a₁. For sequence 10, 7, 4..., d = 7 − 10 = −3, NOT +3.

1. Confusing 'n' with 'aₙ':

• n is the position / term number (1st, 2nd, 3rd, etc.). It must always be a positive whole number (1, 2, 3...).
• aₙ is the value of that term, which can be negative, zero, or a fraction.

1. Brackets with negative numbers:

• Writing 10 + 29 × −3 without brackets often leads to incorrect subtraction. Always write 10 + [29 × (−3)] = 10 − 87.

1. Forgetting to check the pattern:

• A sequence is an AP only if a₂ − a₁ = a₃ − a₂ = a₄ − a₃. Don't check just the first two terms.

Quick Revision Points:

• An AP is defined by just two numbers: its first term a and common difference d.
• If three numbers a, b, c are in AP, then 2b = a + c.
• If Sₙ is given as a formula in terms of n, you can find the first term by putting n = 1 (since S₁ = a₁), and the second term by a₂ = S₂ − S₁.