Fix positive integers $n,j,k$.How many integer sequences are there of the form $1\le a_1<a_2<\ldots<a_k\le n$,where $a_{i+1}-a_i\ge j$ for all $1\le i\le k-1$.
1. **Substitution**: We start by making the substitution \( b_i = a_i - (j - 1)(i - 1) \). This substitution helps to transform the problem into a simpler form. Specifically, we define:
   \[
   b_i = a_i - (j - 1)(i - 1)
   \]
   for \( i = 1, 2, \ldots, k \).

2. **Range Transformation**: We need to determine the range of \( b_i \). Since \( 1 \le a_1 < a_2 < \ldots < a_k \le n \) and \( a_{i+1} - a_i \ge j \), we can rewrite these conditions in terms of \( b_i \):
   \[
   1 \le b_1 < b_2 < \ldots < b_k \le n - (j - 1)(k - 1)
   \]
   This is because:
   \[
   b_i = a_i - (j - 1)(i - 1) \implies a_i = b_i + (j - 1)(i - 1)
   \]
   and the condition \( a_{i+1} - a_i \ge j \) becomes:
   \[
   (b_{i+1} + (j - 1)i) - (b_i + (j - 1)(i - 1)) \ge j \implies b_{i+1} - b_i \ge 1
   \]
   which is always true since \( b_{i+1} > b_i \).

3. **Counting Sequences**: The problem now reduces to counting the number of strictly increasing sequences \( 1 \le b_1 < b_2 < \ldots < b_k \le n - (j - 1)(k - 1) \). This is a classic combinatorial problem where we need to choose \( k \) distinct elements from a set of size \( n - (j - 1)(k - 1) \).

4. **Combinatorial Solution**: The number of ways to choose \( k \) distinct elements from \( n - (j - 1)(k - 1) \) elements is given by the binomial coefficient:
   \[
   \binom{n - (j - 1)(k - 1)}{k}
   \]

Conclusion:
\[
\boxed{\binom{n - (j - 1)(k - 1)}{k}}
\]