2018÷2×(2018+1)÷673 = 3027

They all have a common denominator so we can just combine and simplify the numerators first.


2018² - 2017² + 2016² - 2015² + ... + 2² - 1²


Now, we use the property a² - b² = (a + b)(a - b)


We group these 2018 terms into pairs.

(2018² - 2017²) + (2016² - 2015²) + ... + (2² - 1²)


For each pair, it goes something like this :

2018² - 2017² = (2018 + 2017)(2018 - 2017)
= (2018 + 2017)(1)
= 2018 + 2017

Likewise,


2016² - 2015² = (2016 + 2015)(2016 - 2015)
= (2016 + 2015)(1)
= 2016 + 2015


Similarly, the last pair will be :

2² - 1² = (2 + 1)(2 - 1)
= (2 + 1)(1)
= 2 + 1


So for the entire expression, it simplifies to

2018 + 2017 + 2016 + 2015 + ... + 2 + 1

Next , since there are 2018 terms,

We can pair the 2018 terms like this :

1 + 2018 = 2019
2 + 2017 = 2019
3 + 2016 = 2019
...
...
...
1009 + 1010 = 2019


Each pair adds up to 2019.


For 2018 terms there are 2018 ÷ 2 = 1009 pairs


So the total = 2019 × 1009

Lastly, we need to divide by 673.

Realise that 673 × 3 = 2019
So 2019 ÷ 673 = 3


Therefore , putting them all together,

(2018² - 2017² + 2016² - 2015² + ... + 2² - 1²) / 673

= (2018 + 2017 + 2016 + 2015 + ... + 2 + 1) / 673

= (1 + 2018)(2018 ÷ 2) / 673

= (2019 × 1009) / 673

= 3 × 1009

= 3027