Firstly we try change your recursive dp for linear (and we will pre calculate("precalc" or "pre calc") result in array to find next time by O(1)).

for (int i=1;i<=100000;i++){
        d[i]=d[i-1];
        if (i-k>=0)d[i]+=d[i-k];
        d[i]%=INF;
}    

Now we can find fastly your dp(x) by d[x]
After this we must to find sum of all elements from d[a] to d[b]
we can do it by create a new array that will have
all sum from 1 to i(sum[i]=d[1]+...+d[i]) by precalc

for (int i=1;i<=100000;i++){
    sum[i]=sum[i-1]+d[i];
}

then we know that answer for sum from a to b like this cout<<sum[b]-sum[a-1];
You can look for my code http://codeforces.me/contest/474/submission/9871904