Animated Solution for Mathematics - Sequence and Series: The sum of integers from 1 to 100 that are divisible by 2 or 5 is

To identify the numbers between $1$ and $100$ divisible by $2$ or $5$, we visualize two sets: Set $A$ containing multiples of $2$, and Set $B$ containing multiples of $5$. We seek the sum of the elements in the union of these two sets.

If we simply add the sum of Set $A$ to the sum of Set $B$, we encounter a classic trap. The numbers divisible by both $2$ and $5$ (the intersection) are counted twice.

To correct this, we apply the Principle of Inclusion-Exclusion. We must calculate the sum of the first set, add the sum of the second set, and subtract the sum of the intersection to balance the scales.

A sequence such as $2,4,6,\dots ,100$ is an Arithmetic Progression (A.P.). The sum of an A.P. is governed by the formula:

Here, $n$ is the number of terms, $a$ is the first term, and $l$ is the last term.

For multiples of $2$, we have $a=2$, $l=100$, and $n=\frac{100}{2}=50$. The sum is:

For multiples of $5$, we have $a=5$, $l=100$, and $n=\frac{100}{5}=20$. The sum is:

The numbers divisible by both $2$ and $5$ are the multiples of their Least Common Multiple, which is $10$. These numbers ($10,20,\dots ,100$) were included in both $S_2$ and $S_5$.

We must subtract this intersection once to correct our total. This sequence has $n=\frac{100}{10}=10$ terms.

We now combine these results using the inclusion-exclusion principle: $S_{Total}=S_2+S_5-S_{10}$.

Substituting our calculated values:

The final answer is 3050. This result is derived not by brute force, but by the elegant application of logic and pattern recognition.