# Inter 1st Year Maths 1A Mathematical Induction Formulas

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