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

## TS Inter 1st Year Maths 1A Mathematical Induction Formulas

→ 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 = \(\frac{n(n+1)}{2}\)
- 1
^{2}+ 2^{2}+ 3^{2}+ ……….. + n^{2}= \(\frac{n(n+1)(2 n+1)}{6}\) - 1
^{3}+ 2^{3}+ 3^{3}+ ………… + n^{3}= \(\frac{n^2(n+1)^2}{4}\) - 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) = n
^{2} - Sum of the first n’ even natural numbers : 2 + 4 + 6 + …………….. + (2n) = n (n + 1)