Class 10 / Maths
Class 10 Maths Arithmetic Progressions Notes
Notes on nth term, common difference, and sum of first n terms of an arithmetic progression.
Topic introduction
Arithmetic Progressions is a pattern chapter. Most questions become simple when you identify the first term, common difference, required term number, and whether the question asks for a term or a sum.
Key points
- An arithmetic progression is a list of numbers in which the difference between consecutive terms is constant.
- The nth term of an AP is a_n = a + (n - 1)d.
- The sum of first n terms is S_n = n/2[2a + (n - 1)d].
- If the first and last terms are known, S_n = n/2(a + l).
Definitions
Arithmetic progression: A sequence in which the difference between any two consecutive terms is constant.
First term: The first number in the AP, usually denoted by a.
Common difference: The fixed difference between consecutive terms, usually denoted by d.
nth term: The term at position n in the sequence.
Formulas
- a_n = a + (n - 1)d
- S_n = n/2[2a + (n - 1)d]
- S_n = n/2(a + l), where l is the last term
Examples
- In 3, 7, 11, 15, ... the first term is 3 and common difference is 4.
- For a = 5, d = 3, n = 10, the 10th term is 5 + 9 x 3 = 32.
- The two-digit numbers divisible by 3 form an AP: 12, 15, 18, ..., 99.
Identifying an AP
Check the difference between consecutive terms. If the difference is constant, the sequence is an AP.
The constant difference is called the common difference and is usually denoted by d.
Choosing the right formula
Use a_n = a + (n - 1)d when the question asks for a specific term, number of terms, or common difference.
Use the sum formula when the question asks for total, accumulated value, or sum of a pattern.
Common mistakes
- Using n instead of n - 1 in the nth term formula.
- Taking d as positive even when the AP is decreasing.
- Using the sum formula when the question asks for only one term.
- Forgetting to find the number of terms before calculating a sum.
- Not checking whether a sequence has a constant difference before calling it an AP.
Quick practice
- Find the 20th term of 3, 7, 11, 15, ...
- Find the sum of first 25 terms of 5, 8, 11, 14, ...
- Which term of the AP 7, 13, 19, ... is 97?
- Find d if the 10th term of an AP is 45 and first term is 9.
Related practice
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