Find the sum of all three-digit natural numbers which are divisible by $7$.
Step-by-Step Solution
Key Concept: Identify the first and last three-digit multiples of 7, then apply the sum formula.
Smallest three-digit multiple of $7$: $105$ (since $100/7\approx14.3$, so $15\times7=105$). Largest: $994$ (since $999/7\approx142.7$, so $142\times7=994$). [1.0 Mark]
$a=105,d=7,l=994$. Number of terms: $994=105+(n-1)7\Rightarrow n-1=127\Rightarrow n=128$. [1.0 Mark]
$S_{128}=\dfrac{128}{2}(105+994)=64\times1099=70336$. [1.0 Mark]
Correct Answer: