**ARITHMETIC PROGRESSIONS (AP) — COMPREHENSIVE CHEAT SHEET**
**1. DEFINITION & BASIC CONCEPTS**
• **Arithmetic Progression (AP):** A sequence of numbers where each term (except the first) is obtained by adding a fixed number to the preceding term
• **Common Difference (d):** The fixed number added to each term → d = a_{k+1} − a_k (where a_k is any term)
• **First Term:** Denoted by 'a' (or a₁)
• **General Form of AP:** a, a+d, a+2d, a+3d, ...
• **Key Property:** The difference between any two consecutive terms is constant and equals d
**IDENTIFYING AN AP:**
**EXAMPLE:** For 6, 9, 12, 15: d = 9−6 = 3, difference is constant → AP with a=6, d=3
**EXAMPLE:** For 1, 1, 2, 3, 5: differences are 0, 1, 1, 2 (NOT constant) → NOT an AP
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**2. NATURE OF COMMON DIFFERENCE (d)**
• **d > 0:** AP is increasing (e.g., 2, 4, 6, 8 where d=2)
• **d < 0:** AP is decreasing (e.g., 10, 7, 4, 1 where d=−3)
• **d = 0:** All terms are equal/constant AP (e.g., 5, 5, 5, 5)
• **d can be integer, fraction, decimal, or negative**
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**3. TYPES OF AP**
• **Finite AP:** Has a last term (e.g., 147, 148, 149, ..., 157)
• **Infinite AP:** No last term, continues indefinitely (e.g., 1, 2, 3, 4, ...)
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**4. NTH TERM FORMULA — MOST IMPORTANT**
**Formula:** aₙ = a + (n−1)d
**Where:**
• aₙ = nth term of AP
• a = first term
• d = common difference
• n = position/number of the term
**PROOF CONCEPT:** First term = a, Second term = a+d, Third term = a+2d, ... Pattern shows nth term = a+(n−1)d
**APPLICATIONS:**
**EXAMPLE:** AP is 3, 6, 9, ... Find 10th term: a=3, d=3, a₁₀ = 3+(10−1)×3 = 3+27 = 30
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**5. SUM OF FIRST N TERMS — CRITICAL FORMULA**
**Formula 1:** Sₙ = n/2 [2a + (n−1)d]
**Formula 2:** Sₙ = n/2 [a + l], where l is the last term (aₙ)
**WHICH FORMULA TO USE:**
• Use Formula 1 when you know a, d, n
• Use Formula 2 when you know a, last term l, n
**PROOF CONCEPT:** Write sum forward and backward, add them, simplify → Sₙ = n/2(2a + (n−1)d)
**IMPORTANT RELATIONSHIPS:**
• aₙ = Sₙ − Sₙ₋₁ (for n ≥ 2)
• a₁ = S₁
• If you know Sₙ, you can find individual terms
**EXAMPLE:** Find sum of first 15 terms of AP: 2, 5, 8, ...
a=2, d=3, S₁₅ = 15/2[2(2)+(15−1)3] = 15/2[4+42] = 15/2(46) = 345
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**6. COMMON MISTAKES & HOW TO AVOID**
• **Mistake 1:** Confusing aₙ (the term itself) with Sₙ (sum of n terms) → Remember aₙ is singular term, Sₙ is total sum
• **Mistake 2:** Using wrong formula for d → Always calculate as (next term − previous term), not reversed
• **Mistake 3:** Forgetting (n−1) in nth term formula → The formula is a + (n−1)d, NOT a + nd
• **Mistake 4:** Counting error in position → If first term is a₁, second is a₂, so position n uses (n−1) multiplier
• **Mistake 5:** Sign errors with negative d → Be careful with negative common differences, use parentheses
• **Mistake 6:** Using wrong sum formula → Check what information is given before choosing Formula 1 or 2
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**7. PROBLEM TYPES & STANDARD SOLUTIONS**
**TYPE 1: Identify AP and find a, d**
Solution: Check if consecutive differences are equal → Calculate a (first term) and d (any difference)
**TYPE 2: Find nth term**
Solution: Identify a, d, n → Use aₙ = a+(n−1)d → Calculate
**TYPE 3: Find sum of n terms**
Solution: Identify a, d, n → Use Sₙ = n/2[2a+(n−1)d] OR Sₙ = n/2[a+l] → Calculate
**TYPE 4: Find first term given aₙ, d, n**
Solution: Rearrange aₙ = a+(n−1)d → a = aₙ − (n−1)d → Calculate
**TYPE 5: Find common difference given a, aₙ, n**
Solution: Rearrange aₙ = a+(n−1)d → d = (aₙ−a)/(n−1) → Calculate
**TYPE 6: Word problems (salary, ladder rungs, etc.)**
Solution: Identify first term a and common difference d from context → Translate to AP → Apply nth term or sum formula
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**8. SPECIAL CASES & COROLLARIES**
• **Constant AP (d=0):** All terms equal, Sₙ = n×a
• **Natural Numbers:** 1, 2, 3, ... have a=1, d=1, Sₙ = n(n+1)/2
• **Odd Numbers:** 1, 3, 5, 7, ... have a=1, d=2, aₙ = 2n−1, Sₙ = n²
• **Even Numbers:** 2, 4, 6, 8, ... have a=2, d=2, aₙ = 2n, Sₙ = n(n+1)
• **Arithmetic Mean:** If three terms a, b, c are in AP, then b = (a+c)/2 (middle term is average of extremes)
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**9. FORMULA SUMMARY TABLE**
| **Concept** | **Formula** | **When to Use** |
|---|---|---|
| Common Difference | d = aₖ₊₁ − aₖ | Verifying AP or finding d |
| nth Term | aₙ = a+(n−1)d | Find any specific term |
| Sum of n terms (Type 1) | Sₙ = n/2[2a+(n−1)d] | Know a, d, n |
| Sum of n terms (Type 2) | Sₙ = n/2[a+l] | Know a, last term, n |
| nth from last term | aₙ = l−(n−1)d | Counting from end |
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**10. QUICK REVISION CHECKLIST**
✓ AP definition: constant difference between consecutive terms
✓ Always find a (first term) and d (common difference) first
✓ aₙ formula uses (n−1), not n
✓ Two sum formulas exist—choose based on given information
✓ For word problems, first establish the AP clearly
✓ Check signs carefully, especially with negative d
✓ Verify your AP by checking if d is constant
✓ Sum of n terms can also be written as Sₙ = n/2 × (first term + last term)
Q1. A student observes that in the sequence 8000, 8500, 9000, 9500, ..., each term is obtained by adding 500 to the previous term. Which property of Arithmetic Progressions does this directly illustrate?
Answer: A — The definition of an AP requires that the difference between consecutive terms (common difference d) is constant; in this case d = 500, which is exactly what distinguishes an AP from other sequences.
Q2. Consider the ladder rung lengths: 45, 43, 41, 39, 37, 35, 33, 31. A student claims this is not an AP because the lengths are decreasing. Using the definition of an AP, is this claim correct?
Answer: B — An AP is defined by a constant common difference (here d = –2), not by whether the sequence increases or decreases; negative common differences are explicitly allowed by the AP definition.
Q3. Assertion (A): The sequence 3, 3, 3, 3, ... is an Arithmetic Progression. Reason (R): In this sequence, the common difference d = 0, and the definition of AP permits d to be zero. Choose the correct option:
Answer: A — Both statements are true; R directly explains why A is true because the definition explicitly states d can be any fixed number including zero.
Q4. A student is given the sequence –1.0, –1.5, –2.0, –2.5, ... and claims it cannot be an AP because it contains decimals and negative numbers. Based on the AP definition, evaluate this reasoning.
Answer: B — The AP definition makes no restriction on the type of numbers (integers, decimals, negative numbers are all allowed); here d = –0.5 is constant, confirming it is an AP.
Q5. Assertion (A): To completely determine an AP, you must know both the first term a and the common difference d. Reason (R): Knowing only a without d, or only d without a, does not allow you to write down the sequence uniquely. Choose the correct option:
Answer: A — R directly justifies A: without both parameters, different APs are possible (e.g., a = 6, d = 3 gives 6, 9, 12, ... while a = 6, d = –3 gives 6, 3, 0, ..., both different despite sharing one parameter).
Q6. A teacher writes two sequences on the board: Sequence P: 100, 150, 200, 250, ... and Sequence Q: 12, 22, 32, .... The teacher claims both are APs. Based on the AP definition, which claim is correct?
Answer: B — Sequence P has a constant common difference d = 50, making it an AP; Sequence Q (1, 4, 9, 16, ...) has differences 3, 5, 7, ... which are not constant, so it is not an AP.
Q7. For the general form of an AP: a, a + d, a + 2d, a + 3d, ..., a student writes the nth term as a + (n – 1)d instead of a + nd. Is this correct or incorrect?
Answer: B — The nth term formula a + (n – 1)d is correct because the first term (n = 1) is a + (1 – 1)d = a, the second term (n = 2) is a + (2 – 1)d = a + d, confirming the pattern.
Q8. Assertion (A): The list of heights 147, 148, 149, ..., 157 is a finite Arithmetic Progression. Reason (R): A finite AP has a last term, whereas infinite APs do not have a last term. Choose the correct option:
Answer: A — R directly explains A: the sequence has a clear last term (157) and constant difference (1), making it a finite AP by definition.
Q9. A student claims that the sequence 6, 9, 12, 15, ... proves that knowing only the common difference d = 3 is enough to determine an AP uniquely. Why is this claim incomplete?
Answer: B — Knowing d = 3 alone leaves infinitely many possible APs (3, 6, 9, ..., or 10, 13, 16, ..., etc.); both a and d are needed to uniquely determine the AP.
Q10. Assertion (A): The balance money sequence 950, 900, 850, 800, ..., 50 is an Arithmetic Progression. Reason (R): The common difference is –50, and each term is obtained by subtracting 50 from the previous term. Choose the correct option:
Answer: A — R directly justifies A: the constant difference d = –50 between consecutive terms is the defining property of an AP.
What is an Arithmetic Progression (AP)?
A sequence where each term is obtained by adding a fixed number (common difference d) to the preceding term.
What is the common difference d in an AP?
The fixed number added to each term to get the next term, found by subtracting any term from the term immediately following it.
Write the general form of an AP.
a, a+d, a+2d, a+3d, ... where a is the first term and d is the common difference.
Formula for the nth term of an AP
an = a + (n-1)d, where a is the first term, d is common difference, and n is the position of the term.
How do you identify if a list of numbers is an AP?
Check if the difference between any two consecutive terms is the same; if yes, it's an AP.
Can the common difference d be negative?
Yes, a negative d means the AP is decreasing (each term is smaller than the previous term).
What is a finite AP?
An AP that has a fixed number of terms and a last term (e.g., 45, 43, 41, ..., 31).
What is an infinite AP?
An AP that continues indefinitely without a last term (e.g., 1, 2, 3, 4, ...).
If a = 8 and d = 0, what kind of AP is formed?
A constant AP where all terms are the same: 8, 8, 8, 8, ... (common difference is zero).
What two pieces of information uniquely determine an AP?
The first term a and the common difference d; knowing both allows you to write any term or the entire sequence.
Write the first four terms of the AP where the first term a = 3 and the common difference d = -0.5. Is this AP finite or infinite? [2 marks]
Use the general form a, a+d, a+2d, a+3d to find the four terms. Then explain whether the AP stops at a specific term (finite) or continues forever (infinite) based on whether a last term is mentioned.
Reena's monthly salary starts at ₹8000 with an annual increment of ₹500. (a) How much will her salary be in the 5th year? (b) Write the AP representing her salary for the first 5 years. [3 marks]
Here a = 8000 and d = 500 (annual increment). Use the formula an = a + (n-1)d to find the 5th year salary. For part (b), list the terms: a, a+d, a+2d, a+3d, a+4d to show the AP for 5 years.
The ladder in Example (ii) has rungs whose lengths decrease uniformly by 2 cm from bottom to top, with the bottom rung being 45 cm. (a) Find the length of the 8th rung from the bottom. (b) Find the sum of the lengths of all 8 rungs. (c) Is this AP finite or infinite, and why? [5 marks]
Identify a = 45 (bottom rung), d = -2 (decrease), and n = 8. Use an = a + (n-1)d to find the 8th rung. For the sum, use Sn = n/2[2a + (n-1)d] or Sn = n/2(first + last). For part (c), explain that since the ladder has exactly 8 rungs, the AP has a fixed last term, making it finite.
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