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 = \(\frac{n(n+1)}{2}\)
- Σn2 = \(\frac{n(n+1)(2 n+1)}{6}\)
- Σn3 = \(\frac{n^{2}(n+1)^{2}}{4}\)
- a, (a + d), (a + 2d), ……….. are in a.p
n th term tn = a + (n – 1)d, sum of n terms Sn = \(\frac{n}{2}\)[ 2a + (n – 1)d] = \(\frac{n}{2}\)[a + l]
a = first term, l= last term. - a, ar, ar2, ………… is a g.p
Nth terms tn = a.rn-1 a = 1st term, r = common ratio - Sum of n terms sn = a\(\frac{\left(r^{n}-1\right)}{r-1}\); r > 1 = a\(\left(\frac{1-r^{n}}{1-r}\right)\); r < 1