Question

P = \(2\times3+3\times4+4\times5+5\times6+...+99\times100\)

Answer 1

P = 2 x 3 + 3 x 4 + ...+ 99 x 100

=> 3 x P = 2 x 3 x 3 + 3 x 4 x 3 + ....+ 99 x 100 x 3

3 x P = 2 x 3 x ( 4-1) + 3 x 4 x (5-2) + ...+ 99 x 100 x ( 101 -98)

3 x P = 2 x 3 x 4 - 1 x 2 x 3 + 3 x 4  x  5- 2 x 3 x 4 + ...+ 99 x 100 x 101 - 98 x 99 x 101

3 x P = ( 2  x 3  x 4 + 3 x 4 x  5 + ...+ 99 x 100 x 101) - ( 1 x 2 x 3 + 2 x 3  x 4 + ...+ 98 x 99 x 101)

3 x P = 99  x 100 x 101 - 1 x 2  x 3

\(P=\frac{99x100x101-1x2x3}{3}=333298\)

Answer 2

p=2.3+3.4+4.5+5.6+...+99.100

3p=2.3.3+3.4.3+4.5.3+5.6.3+...+99.100.3

3p=2.3.(4-1)+3.4.(5-2)+4.5.(6-3)+5.6.(7-4)+...+99.100.(101-98)

3p=2.3.4-1.2.3+3.4.5-2.3.4+4.5.6-3.4.5+5.6.7-4.5.6+99.100.101-98.99.100

3p=98.99.100-1.2.3

p=\(\frac{98.99.100-1.2.3}{3}=323398\)