Question

prove by mathematical induction that
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Answer 1

To prove that :

$\sf \: p(n) : 1.2 + 2.3 + 3.4 + \cdots \: n(n + 1) = \dfrac{ n(n + 1)(n + 2)}{3} for \: all \: natural \: numbers$

Let's check the validity of the sequence for n = 1.

LHS : 1 × 2 = 2

RHS : 1(1+1)(1+2)/3 = 1 × 2 = 2

True for P(1).

Now, let us assume that the sequence holds good for P(k) for all values of k, where k belongs to natural numbers.

$\sf \: p(k) : 1.2 + 2.3 + 3.4 + \cdots \: k(k + 1) = \dfrac{ k(k + 1)(k + 2)}{3} - - - - - - - (1)$

We have to prove that the sequence is valid for P(k + 1), where k + 1 belongs to natural numbers.

$\sf \: p(k + 1) : 1.2 + 2.3 + 3.4 + \cdots \: (k + 1)(k + 2) = \dfrac{ (k + 1)(k + 2)(k + 3)}{3}$

LHS.

$\sf \: 1.2 + 2.3 + 3.4 + \cdots \: (k + 1)(k + 2) \\ \\ \longrightarrow \sf \: \{1.2 + 2.3 + 3.4 \cdots\} + (k + 1)(k + 2) \: \\ \\ \longrightarrow \sf \: \dfrac{ k(k + 1)(k + 2)}{3} + (k + 1)(k + 2) \: \: \: |from \: (1)| \\ \\ \longrightarrow \sf \:(k + 1)(k + 2) \bigg[ \dfrac{ k}{3} + 1 \bigg] \\ \\ \longrightarrow \sf\dfrac{ (k + 1)(k + 2)(k + 3)}{3}$

Hence, proved.

By Principle of Mathematical Induction, P(n) is true for all natural numbers.