11. An infinite arithmetic progression of positive integers contains the terms $7,11,15,71,75$ and 79. The first term in the progression is 7. Kim writes down all the possible values of the onehundredth term in the progression. What is the sum of the numbers Kim writes down?
11. 714 The difference between any two terms is either 4 or is a multiple of 4 . So the term-toterm difference in the progression must be a divisor of 4 . Since all the terms of the progression are integers the only feasible differences are 1,2 and 4 .
If the term-to-term difference is 1 then the one-hundredth term will be $7+99 \times 1=106$. If the term-to-term difference is 2 then the one-hundredth term will be $7+99 \times 2=205$. If the term-to-term difference is 4 then the one-hundredth term will be $7+99 \times 4=403$. The sum of these numbers is 714 .