Question

$P\left(n\right):1.2+2.3+3.4+...+n(n+1)=\frac{(n+1)(n+2)}{3}$

The statement P(n) is

• A. is true for all natural numbers
• B. is not true for all natural numbers
• C. is true for n=1
• D. is true for n=2

Answer 1

The correct option is B

is not true for all natural numbers

$P\left(n\right):1.2+2.3+3.4+...+n(n+1)=\frac{(n+1)(n+2)}{3}P\left(1\right):1.2=\frac{2.3}{3}\Rightarrow 2=2-true$
Assume P(k) is true
$\Rightarrow 1.2+2.3+3.4+...+k(k+1)=\frac{(k+1)(k+2)}{3}Now,P(k+1):1.2+2.3+3.4+...+k(k+1)+(k+1)(k+2)=\frac{(k+1)(k+2)}{3}+(k+1)(k+2)=\frac{4(k+1)(k+2)}{3}\ne \frac{(k+2)(k+3)}{3}$

P(k+1) is not true.
Hence, P(n) is not true for all natural numbers