TS Inter 1st Year Maths 1A Mathematical Induction Formulas

Learning these TS Inter 1st Year Maths 1A Formulas Chapter 2 Mathematical Induction will help students to solve mathematical problems quickly.

→ Principle of finite Mathematical Induction : Let S(n) be a statement of a result for each n ∈ N. If

S(1) is true
S(K) is true ⇒ S(K + 1) is also true then S(n) is true ∀ n ∈ N. (Set of natural numbers = N).

→ Principle of complete Mathematical Induction : Let S(n) be a statement for each n ∈ N. If

S(T) is true
S(1), S(2), S(3), ……….. S(K) are true ⇒ S(K + 1) is true, then S(n) is true, ∀ n ∈ N.

→ Useful formulae:

1 + 2 + 3 + ………. + n = \(\frac{n(n+1)}{2}\)
1² + 2² + 3² + ……….. + n² = \(\frac{n(n+1)(2 n+1)}{6}\)
1³ + 2³ + 3³ + ………… + n³ = \(\frac{n^2(n+1)^2}{4}\)
The nth term of the arithmetic progression (A.P.) is t_n = a + (n – 1) d
The sum f n terms of the arithmetic progression (A.P.) is S_n = \(\frac{n}{2}\) [2a + (n – 1) d]
The nth term of the geometric progression (G.P.) is t_n = a. r^n-1
The sum of the n terms in G.P is S_n = \(\frac{a\left(r^n-1\right)}{r-1}\). r > 1
Sum of the first n’ odd natural numbers : 1 + 3 + 5 + ……………….. + (2n – 1) = n²
Sum of the first n’ even natural numbers : 2 + 4 + 6 + …………….. + (2n) = n (n + 1)