A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
Arithmetic Progressions - Quick Look Revision Guide
Your 1-page summary of the most exam-relevant takeaways from Mathematics.
This compact guide covers 20 must-know concepts from Arithmetic Progressions aligned with Class X preparation for Mathematics. Ideal for last-minute revision or daily review.
Complete study summary
Essential formulas, key terms, and important concepts for quick reference and revision.
Key Points
Define AP with an example.
An Arithmetic Progression (AP) is a sequence where each term after the first is obtained by adding a fixed number, called the common difference (d). Example: 2, 5, 8, 11, ... where d=3.
General form of AP.
The general form of an AP is: a, a+d, a+2d, a+3d, ..., where 'a' is the first term and 'd' is the common difference.
nth term formula.
The nth term of an AP is given by: a_n = a + (n-1)d, where 'a' is the first term, 'd' is the common difference, and 'n' is the term number.
Sum of first n terms.
The sum of the first n terms (S_n) of an AP is: S_n = n/2 [2a + (n-1)d] or S_n = n/2 (a + l), where 'l' is the last term.
Common difference calculation.
The common difference (d) can be found by subtracting any term from the succeeding term: d = a_{k+1} - a_k.
Finite vs Infinite AP.
An AP with a last term is finite (e.g., 10, 15, 20, ..., 100). An AP without a last term is infinite (e.g., 3, 6, 9, 12, ...).
Real-world AP example.
Salary increment: Starting salary ₹8000, annual increment ₹500 forms an AP: 8000, 8500, 9000, ...
Negative common difference.
If d is negative, the AP decreases. Example: 15, 12, 9, 6, ... where d=-3.
Zero common difference.
If d=0, all terms are equal. Example: 7, 7, 7, 7, ... is an AP with d=0.
Finding number of terms.
Use the nth term formula to find 'n' when the last term is known. Example: For AP 5, 9, 13, ..., 45, find n using a_n = a + (n-1)d.
AP in patterns.
APs are used in patterns like rungs of a ladder decreasing uniformly, or petals of a sunflower following a fixed increment.
Mid-term in AP.
For three consecutive terms in AP, the middle term is the arithmetic mean of its neighbors. If a, b, c are in AP, then b = (a + c)/2.
Sum of natural numbers.
Sum of first n natural numbers is an AP sum: S_n = n(n+1)/2. Example: Sum of first 100 numbers is 5050.
AP in finance.
Simple interest calculation over years forms an AP where each term increases by a fixed amount based on the principal and rate.
Misconception: All sequences are APs.
Not all sequences are APs. Only those with a constant difference between consecutive terms qualify as APs.
Memory hack for nth term.
Remember 'a_n = a + (n-1)d' as 'First term + (Term number - 1) × Common difference'.
Sum of terms from end.
To find the sum of terms from the end, reverse the AP and use the same sum formula.
AP in geometry.
Number of unit squares in squares with sides 1, 2, 3, ... units forms an AP of squares: 1, 4, 9, 16, ...
Finding missing terms.
If some terms are missing in an AP, use the common difference to find them. Example: Given 3, _, 7, the missing term is 5.
AP in daily life.
APs model many real-life scenarios like saving money regularly, annual salary increments, or decreasing lengths of ladder rungs.
Real Numbers encompass all rational and irrational numbers, forming a complete and continuous number line essential for various mathematical concepts.
Explore the world of Polynomials, understanding their types, degrees, and operations to solve algebraic expressions and equations effectively.
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
