Use these Inter 1st Year Maths 1A Formulas PDF Chapter 2 Mathematical Induction to solve questions creatively.

## Intermediate 1st Year Maths 1A Mathematical Induction Formulas

**Principle of finite mathematical induction:**

Let S be a subset of N such that

- 1 ∈ S
- For any k ∈ N, k ∈S ⇒ (k + 1) ∈ S

Then S = N

**Principle of complete mathematical induction:**

Let S be a subset of N such that

- 1 ∈ S
- For any k ∈ N {1, 2, 3 … k} ⊆ S

⇒ (k + 1) ∈ S

Then S = N

**Steps to prove a statement using the principle of mathematical induction :**

- Basis of induction : Show that P(1) is true
- Inductive hypothesis : For k > 1, assume that P(k) is true
- Inductive Step : Show that P(k + 1) is true on the basis of the inductive hypothesis.

**Principle of finite Mathematical Induction:**

Let {P(n) / n ∈ N} be a set of statements. If

- p(1) is true
- p (m) is true ⇒ p (m+1) is true ; then p (n) is true for every n ∈ N.

**Principle of complete induction:**

Let {P (n) / n N} be a set of statements. If p (1) is true and p(2), p(3) …. p (m-1) are true ⇒ p(m) is true, then p (n) is true for every n e N.

Note:

- The principle of mathematical induction is a method of proof of a statement.
- We often use the finite mathematical induction, hence or otherwise specified the mathematical induction is the finite mathematical induction.

**Some important formula:**

- Σn = \(\frac{n(n+1)}{2}\)
- Σn
^{2}= \(\frac{n(n+1)(2 n+1)}{6}\) - Σn
^{3}= \(\frac{n^{2}(n+1)^{2}}{4}\) - a, (a + d), (a + 2d), ……….. are in a.p

n th term t_{n}= a + (n – 1)d, sum of n terms S_{n}= \(\frac{n}{2}\)[ 2a + (n – 1)d] = \(\frac{n}{2}\)[a + l]

a = first term, l= last term. - a, ar, ar
^{2}, ………… is a g.p

Nth terms t_{n}= a.r^{n-1}a = 1st term, r = common ratio - Sum of n terms s
_{n}= a\(\frac{\left(r^{n}-1\right)}{r-1}\); r > 1 = a\(\left(\frac{1-r^{n}}{1-r}\right)\); r < 1