What is Arithmetic Progression (A.P.)?
An Arithmetic Progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the common difference (d). The general form of an A.P. is:
a₁, a₂, a₃, ..., an where each term is obtained by adding the common difference to the previous term.
Key Components:
First term (a₁): The first number in the sequence.
Common Difference (d): The fixed amount added to each term to get the next term.
nth term (aₙ): The formula to find the nth term of an A.P. is:
aₙ = a₁ + (n-1)d
where:
aₙ = nth term
a₁ = first term
d = common difference
n = position of the term
Sum of the first n terms (Sₙ): The sum of the terms up to the nth term in an A.P. can be calculated using:
Sₙ = n/2 [2a₁ + (n-1)d]
or
Sₙ = n/2 (a₁ + aₙ)
Example:
Consider the A.P. 3, 7, 11, 15, 19, ...
First term (a₁) = 3
Common difference (d) = 4
nth term (aₙ) can be found using the formula:
aₙ = 3 + (n-1)(4)
Why is A.P. Important?
Arithmetic Progression is an essential concept in mathematics, used in various real-life applications like calculating interest rates, distances, time intervals, and much more. Understanding A.P. helps build a foundation for higher-level mathematical concepts, including sequences, series, and calculus.
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