Find the sum of all two-digit positive integers that are multiples of $7$.

Step 1: The two-digit multiples of $7$ form an Arithmetic Progression (AP):

• Smallest two-digit multiple: $14$ ($7\times 2$)

• Largest two-digit multiple: $98$ ($7\times 14$)

Step 2: The terms are: $14,21,28,\dots ,98$.
The number of terms $n$ is given by:

Step 3: The sum of an AP is given by:

where $a=14$ (first term) and $l=98$ (last term). Substituting the values:

So the sum is $728$.