Animated Solution for Mathematics - Sequence and Series: Let $A=\{x\in \mathbb{R}:[x+3]+[x+4]\le 3\},B=\{x\in \mathbb{R}:3^x{\left({\sum }_{r=1}^{\infty }\frac{3}{10^r}\right)}^{x-3}<3^{-3x}\}$, where $[t]$ denotes greatest integer function. Then,

Imagine you are standing on the vast, infinite number line of mathematics. Sometimes, two paths look completely different—one winding through the forests of step functions, the other climbing the mountains of infinite series—yet they both lead to the exact same destination.

Today, we are going to explore such a journey. We are tasked with comparing two sets, $A$ and $B$, defined by seemingly disparate mathematical structures.

The GIF Mystery

Set $A$ is defined by the inequality $[x+3]+[x+4]\le 3$. At first glance, the Greatest Integer Function (GIF) can feel intimidating.

We have a powerful tool in our arsenal: the property $[x+n]=[x]+n$, where $n$ is any integer. By applying this, we can pull the constants out of the brackets:

Substituting these back, our inequality transforms into:

Combining the like terms, we get $2[x]+7\le 3$. Subtracting $7$ from both sides, we arrive at $2[x]\le -4$, which simplifies to:

Now, pause and visualize the number line. If the greatest integer less than or equal to $x$ is at most $-2$, then $x$ must be strictly less than $-1$.

Therefore, Set $A$ is the interval $(-\infty ,-1)$.

The Infinite Series Challenge

Now, let us turn our attention to Set $B$. It involves an infinite summation:

Focus on the infinite series ${\sum }_{r=1}^{\infty }\frac{3}{10^r}$. This is a classic infinite Geometric Progression where the first term $a=\frac{3}{10}$ and the common ratio $r=\frac{1}{10}$.

Using the sum formula $S_{\infty }=\frac{a}{1-r}$, we calculate:

Substituting this back, our inequality becomes:

By rewriting $\frac{1}{3}$ as $3^{-1}$, we get $3^x(3^{-1}{)}^{x-3}<3^{-3x}$. Using the laws of exponents, this simplifies to:

The Convergence

We are at the final stretch. We have $3^3<3^{-3x}$. Since the base $3$ is greater than $1$, we can compare the exponents directly:

Dividing by $-3$ is the moment of truth—remember to flip the inequality sign! We get $-1>x$, or $x<-1$.

Thus, Set $B$ is also the interval $(-\infty ,-1)$.

It is truly fascinating that we started with a step function and an infinite series, and through the rigor of algebra, we discovered that $A=B$. This problem teaches us that beneath the surface of complex notation, there is often a simple, elegant truth waiting to be uncovered.