Answer 1

(50² – 49²) + (48² – 47²) + (46² – 45²) + ... + (2² – 1²)

= (50² + 48² + 46² + … + 2²) – (49² +  47² + 45² +  …+ 1²)

= `sum_("r" = 1)^25 (2"r")^2 - sum_("r" = 1)^25(2"r" - 1)^2`

= `sum_("r" = 1)^25 4"r"^2 - sum_("r" = 1)^25(4"r"^2 - 4"r" + 1)`

= `sum_("r" = 1)^25[4"r"^2 - (4"r"^2 - 4"r" + 1)]`

= `sum_("r" = 1)^25(4"r" - 1)`

= `4sum_("r" = 1)^25 "r" - sum_("r" = 1)^25 1`

= `4 xx (25(25 + 1))/2 - 25`

= `(4(25)(26))/2 - 25`

= 1300 – 25

= 1275