Notes of Ch 5 Arithmetic Progression| Class 10th Maths

Study Materials and Revision Notes for Ch 5 Arithmetic Progression Class 10th Maths

Arithmetic Progression

Arithmetic Progression: An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number d to the preceding term, except the first term.

Term: The fixed number d is called the common difference of the A.P.

Common Difference: Each number in the list of arithmetic progression is called term. It can be positive, negative or zero.

General form of an AP: If a is first term and d is common difference of an A.P. The general form of an AP is: a, a+d, a+2d, a+3d
(iii) ax3 + bx+ cx + d is  a polynomial of degree 3 cubic polynomial.

Finding General Term of an AP: We can find the nth term of an A.P by using Tn = a + (n-1)d
We can find the required term by putting the numeric value of n(no. of term) which we have to find from starting.
nth term from the end: l - (n-1)d where, l is the last term

Zero of a polynomial: A real number k is said to be zero of a polynomial p(x) if p(k) = 0.
Example: -3/2 is called zero of a polynomial p(x) = 2x + 3 because p(-3/2) = 2x + 3.
(i) A linear polynomial has at most one zero.
(ii) A Quadratic polynomial has at most two zeroes.
(iii) A Cubic polynomial has at most three zeroes.
(iv) A polynomial of degree n has at most n zeroes.

For quadratic polynomial: If α,β are zeroes of polynomial p(x) = ax+ bx + c then:
(i) Sum of zeroes = α + β = -b/a = (-coefficient of x)/(coefficient of x2)
(ii) Product of zeroes = α.β = c/a = (constant term)/(coefficient of x2)
(iii) A quadratic polynomial whose zeroes are α and β, is given by:
p(x) = k[x2 - (α+β)x + αβ] where k is any real number.

For cubic polynomial: If α,β and γ are zeroes of polynomial p(x) = ax3 + bx+ cx + d then:
(i) α + β + γ = -b/a = (-coefficient of x2)/(coefficient of x3)
(ii) αβ + βγ + γα = c/a = (constant term of x)/(coefficient of x3)
(iii) α.β.γ = -d/a = (-constant term)/(coefficient of x3)
(iv) A cubic polynomial whose zeroes are α, β and γ, is given by:
p(x) = k[x3 - (α+β+γ)x2 + (αβ+βγ+γα)x - αβγ] where k is any real number.

Division Algorithm: If p(x) and g(x) are any two polynomials where g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that:
p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).

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