Chapter 5: Arithmetic Progressions
Understanding Arithmetic Progressions (APs)
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This fixed difference is called the common difference (denoted as d). Examples include sequences like 2, 5, 8, 11,… where the common difference is 3. The general form of an AP is:
a,a+d,a+2d,a+3d,…where a is the first term.
APs are used to model regular growth or decline in real-life situations like salary increases or stair-step patterns.
Finding Terms and Sums in an AP
The nth term (or general term) of an AP is given by the formula:
an=a+(n−1)d
This helps in finding any term in the sequence without listing all the previous ones.
To calculate the sum of the first n terms, the formula used is:
Sn=n2[2a+(n−1)d]
or
Sn=n2(a+l) where a is the first term and l is the last term.
These formulas simplify complex calculations involving patterns in real life or examination settings.
Applications and Problem Solving
APs are widely used in solving problems involving daily patterns like saving plans, distributing items in regular intervals, or total distances in repeated movements. The chapter provides multiple examples and exercises to develop skill in forming and solving AP problems using the given formulas. Visual examples and contextual questions help strengthen the understanding of sequences and their real-world relevance.