This chapter introduces arithmetic progressions, which are sequences of numbers generated by adding a fixed value to the previous term. Understanding these patterns is crucial for solving real-life mathematical problems.
Arithmetic Progressions – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class X in Mathematics.
This one-pager compiles key formulas and equations from the Arithmetic Progressions chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
nth term of an AP: aₙ = a + (n - 1)d
aₙ is the nth term, a is the first term, d is the common difference, and n is the term number. This formula helps find any term in an AP. Example: For AP 2, 5, 8,..., the 4th term is 2 + (4-1)*3 = 11.
Sum of first n terms: Sₙ = n/2 [2a + (n - 1)d]
Sₙ is the sum of first n terms, a is the first term, d is the common difference. It calculates the total of first n terms. Tip: Also expressible as Sₙ = n/2 (a + l), where l is the last term.
Common difference: d = aₙ - aₙ₋₁
d is the common difference, aₙ is the nth term, aₙ₋₁ is the (n-1)th term. It finds the difference between consecutive terms. Example: In 3, 7, 11,..., d = 7 - 3 = 4.
Number of terms: n = [(l - a)/d] + 1
n is the number of terms, l is the last term, a is the first term, d is the common difference. Useful when first term, last term, and common difference are known.
Arithmetic Mean: AM = (a + b)/2
AM is the arithmetic mean between two numbers a and b. It’s the middle term in an AP of three terms. Example: AM of 4 and 6 is (4+6)/2 = 5.
Sum when first and last terms are known: Sₙ = n/2 (a + l)
Sₙ is the sum of first n terms, a is the first term, l is the last term. Simplifies calculation when the last term is known.
General form of an AP: a, a + d, a + 2d, ..., a + (n-1)d
Represents the sequence of an AP where each term increases by a common difference d. Example: 5, 9, 13,... where a=5, d=4.
Difference of sums: Sₙ - Sₙ₋₁ = aₙ
The difference between the sum of first n terms and first (n-1) terms gives the nth term. Useful for verifying terms.
Sum of natural numbers: Sₙ = n(n + 1)/2
Special case of AP sum where a=1, d=1. Example: Sum of first 5 natural numbers is 5*6/2 = 15.
Condition for three terms to be in AP: 2b = a + c
For three terms a, b, c to be in AP, twice the middle term must equal the sum of the first and last terms. Example: 3, 7, 11 are in AP as 2*7 = 3 + 11.
Equations
Finding d: d = (aₙ - a₁)/(n - 1)
Calculates common difference d using the first term a₁, nth term aₙ, and number of terms n. Example: For AP with a₁=3, a₅=11, d=(11-3)/(5-1)=2.
Finding n: n = 1 + (aₙ - a)/d
Determines the term number n using the nth term aₙ, first term a, and common difference d. Example: For a=2, d=3, aₙ=14, n=1+(14-2)/3=5.
Sum of terms from mth to nth: S = Sₙ - Sₘ₋₁
Calculates the sum of terms from the mth to the nth term by subtracting the sum up to (m-1)th term from the sum up to nth term.
Last term from sum: l = (2Sₙ)/n - a
Finds the last term l when sum Sₙ, number of terms n, and first term a are known. Rearranged from Sₙ = n/2 (a + l).
First term from sum: a = (2Sₙ)/n - l
Finds the first term a when sum Sₙ, number of terms n, and last term l are known. Derived from Sₙ = n/2 (a + l).
Middle term in odd number of terms: aₘ = Sₙ/n
For an AP with odd number of terms n, the middle term is the average of the sum. Example: For AP 4,7,10,13,16, S₅=50, middle term a₃=50/5=10.
Sum of first n odd numbers: Sₙ = n²
Special case where the sum of first n odd numbers (1,3,5,...) equals n². Example: Sum of first 3 odd numbers is 1+3+5=9=3².
Sum of first n even numbers: Sₙ = n(n + 1)
Special case where the sum of first n even numbers (2,4,6,...) equals n(n+1). Example: Sum of first 4 even numbers is 2+4+6+8=20=4*5.
Product of equidistant terms in finite AP: a₁ * aₙ = a₂ * aₙ₋₁ = ...
In a finite AP, the product of terms equidistant from the start and end are equal. Useful for problems involving product of terms.
Condition for AP: aₖ₊₁ - aₖ = constant
A sequence is an AP if the difference between consecutive terms is constant. This is the defining property of an AP.
This chapter explores real numbers, focusing on key properties such as the Fundamental Theorem of Arithmetic and the concept of irrational numbers, which are crucial for understanding the number system.
This chapter discusses polynomials, their degrees, and classifications such as linear, quadratic, and cubic. Understanding polynomials is essential for solving various mathematical problems.
This chapter focuses on solving pairs of linear equations with two variables and their real-life applications.
This chapter explores quadratic equations, highlighting their forms and significance in real-world applications.
This chapter focuses on the properties of triangles, specifically their similarity and how it can be applied in various real-world contexts.
This chapter covers the concepts of coordinate geometry, including finding distances between points and dividing line segments. Understanding these concepts is essential for solving geometry problems using algebra.
This chapter focuses on the foundational concepts of trigonometry, particularly the relationships between the angles and sides of right triangles.
This chapter explores how trigonometry is applied in real-life situations, particularly in measuring heights and distances.
This chapter explores the properties of circles, particularly focusing on tangents and their relationship with radii and secants.
This chapter focuses on sectors and segments of circles, essential concepts in geometry. Understanding these helps in solving real-life problems related to areas and measurements.
