The nth term of an arithmetic progression (AP) is given by aₙ = a + (n - 1)d, where an is the first term and d is the common difference.
The sum of the first n terms of an AP is given by Sₙ = n/2 [2a + (n - 1)d].
The sum of an AP with first term a common difference d, and number of terms n is also given by Sₙ = n/2 [a + l], where l is the last term.
The last term of an AP is given by l = a + (n - 1)d.
The nth term of an AP from the end is given by aₙ = l - (n - 1)d, where l is the last term.

🪤 The 5 Mistakes That Cost Marks

Not checking if the given sequence is an arithmetic progression by verifying if the difference between consecutive terms is constant.
Not using the correct formula for the nth term or the sum of the first n terms of an arithmetic progression.
Forgetting to subtract 1 from n when using the formula aₙ = a + (n - 1)d to find the nth term.
Not using the formula Sₙ = n/2 [a + l] when the first and last terms are given.
Not verifying the units and dimensions of the answer, especially when the problem involves real-world applications.

✏️ 3 Solved PYQs

Question 1: Find the 10th term of the AP 2, 5, 8, 11, ...
Step 1: Identify the first term (a) and the common difference (d) of the AP: a = 2, d = 5 - 2 = 3.
Step 2: Use the formula aₙ = a + (n - 1)d to find the 10th term: a₁₀ = 2 + (10 - 1)3 = 2 + 9*3 = 2 + 27 = 29.
Answer: 29
Question 2: The sum of the first 50 terms of an AP is 650. If the first term is 5, find the common difference.
Step 1: Use the formula Sₙ = n/2 [2a + (n - 1)d] and substitute the given values: 650 = 50/2 [2*5 + (50 - 1)d].
Step 2: Simplify the equation: 650 = 25 [10 + 49d].
Step 3: Solve for d: 650 = 250 + 1225d, 400 = 1225d, d = 400/1225 = 8/25.
Answer: 8/25
Question 3: Find the sum of the first 20 terms of the AP 3, 7, 11, 15, ...
Step 1: Identify the first term (a) and the common difference (d) of the AP: a = 3, d = 7 - 3 = 4.
Step 2: Use the formula Sₙ = n/2 [2a + (n - 1)d] to find the sum: S₂₀ = 20/2 [23 + (20 - 1)4] = 10 [6 + 194] = 10 [6 + 76] = 10 * 82 = 820.
Answer: 820

🧠 The One Thing Most Students Get Wrong

Most students forget to check if the given sequence is an arithmetic progression by verifying if the difference between consecutive terms is constant, which can lead to incorrect answers.

👁️ Ayush's Note

To find the nth term of an AP, use the formula aₙ = a + (n - 1)d, and to find the sum of the first n terms, use the formula Sₙ = n/2 [2a + (n - 1)d] or Sₙ = n/2 [a + l].
Always verify the units and dimensions of the answer, especially when the problem involves real-world applications.

🔁 Last 5 Minutes Box

Revise the formulas for the nth term and the sum of the first n terms of an arithmetic progression.
Practice finding the nth term and the sum of the first n terms of an AP.
Check if the given sequence is an arithmetic progression by verifying if the difference between consecutive terms is constant.

📝 Practice MCQs

1. The 5th term of the AP 2, 5, 8, 11, ... is

A) 11

B) 14

C) 15

D) 17

Answer: B) 14. Explanation: Use the formula aₙ = a + (n - 1)d to find the 5th term: a₅ = 2 + (5 - 1)3 = 2 + 4*3 = 2 + 12 = 14.

2. The sum of the first 10 terms of the AP 1, 3, 5, 7, ... is

A) 100

B) 110

C) 125

D) 150

Answer: C) 100. Explanation: Use the formula Sₙ = n/2 [2a + (n - 1)d] to find the sum: S₁₀ = 10/2 [21 + (10 - 1)2] = 5 [2 + 92] = 5 [2 + 18] = 5 * 20 = 100.

3. The first term of the AP 7, 10, 13, 16, ... is

A) 3

B) 5

C) 7

D) 10

Answer: C) 7. Explanation: The first term of the AP is given as 7.

4. The common difference of the AP 2, 5, 8, 11, ... is

A) 2

B) 3

C) 4

D) 5

Answer: B) 3. Explanation: The common difference is the difference between consecutive terms: d = 5 - 2 = 3.

5. The 20th term of the AP 3, 7, 11, 15, ... is

A) 73

B) 75

C) 77

D) 79

Answer: C) 77. Explanation: Use the formula aₙ = a + (n - 1)d to find the 20th term: a₂₀ = 3 + (20 - 1)4 = 3 + 19*4 = 3 + 76 = 79.

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This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides:

The nth term of an arithmetic progression (AP) is given by aₙ = a + (n - 1)d, where an is the first term and d is the common difference.
The sum of the first n terms of an AP is given by Sₙ = n/2 [2a + (n - 1)d].
The sum of an AP with first term a common difference d, and number of terms n is also given by Sₙ = n/2 [a + l], where l is the last term.
The last term of an AP is given by l = a + (n - 1)d.
The nth term of an AP from the end is given by aₙ = l - (n - 1)d, where l is the last term.