Question

( 1.2 + 2.3 + 3.4 + ....... + 2012.2013) - ( 2² + 3² + 4² + 5² ........ + 2013² )

Answer 1

(1.2 + 2.3 + 3.4 + ... + 2012.2013) - (2² + 3² + 4² + 5² + ... + 2013²)

= [(2 - 1).2 + (3 - 1).3 + (4 - 1).4 + ... + (2013 - 1).2013] - (2² + 3² + 4² + 5² + ... + 2013²)

= (2² - 2 + 3² - 3 + 4² - 4 + ... + 2013² - 2013) - (2² + 3² + 4² + 5² + ... + 2013²)

= 2² - 2 + 3² - 3 + 4² - 4 + ... + 2013² - 2013 - 2² - 3² - 4² - 5² - ... - 2013²

= (2² - 2²) + (3² - 3²) + (4² - 4²) + ... + (2013² - 2013²) - (2 + 3 + 4 + ... + 2013)

= 0 - (2 + 3 + 4 + ... + 2013)

= 0 - (1 + 2 + 3 + 4 + ... + 2013) + 1

= 0 - \(\dfrac{2013.\left(2013+1\right)}{2}\) + 1

= 0 - 2027091 + 1

= (-2027091) + 1

= -2027090