Question

Question ▶

1.) 1×2+2×3+3×4--------- +n(n+1)=
[n(n+1)(n+2)/3]


2.) For all
$n \geqslant 1 \: prove \: that \:$

Answer 1

Hlw mate!!
1st
Solution:-

LHS = (1) (2) = 2

RHS .=

(1)(1+1)(1+2)3(1)(1+1)(1+2)3

Which is Equal to 2

Assume N = K

(1)(2)+(2)(3)+(3)(4)+⋯+(k)(k+1)=(k)(k+1)(k+2)3(1)(2)+(2)(3)+(3)(4)+⋯+(k)(k+1)=(k)(k+1)(k+2)3

Proof that the equation is true for N = K + 1

(1)(2)+(2)(3)+(3)(4)+⋯+(k)(k+1)+(k+1)(k+2)(1)(2)+(2)(3)+(3)(4)+⋯+(k)(k+1)+(k+1)(k+2)

Which is Equal To:

(k)(k+1)(k+2)3+(k+1)(k+2)(k)(k+1)(k+2)3+(k+1)(k+2)

This is where I've went so far

If I did the calculation right the Answer should be

(k+1)(k+2)(k+3)3(k+1)(k+2)(k+3)3.


Hope it helpful

Answer 2

#RAM RAM ji ❤

2.) well your statement is wrong

Correct statement is For all

$n \geqslant 1 \\ {1}^{2} + {2}^{2} + {3}^{2} + {4}^{2} ... \frac{n(n + 1)(2n + 1)}{6}$

for solution see the attachment I hope it will help you^_^

Thanks ❤

#Nishu HarYanvi ♥