Animated Solution for Mathematics - Complex Numbers: If the set $R=\{(a,b):a+5b=42,a,b\in \mathbb{N}\}$ has $m$ elements and ${\sum }_{n=1}^m(1-i^{n!})=x+iy$, where $i=\sqrt{-1}$, then the value of $m+x+y$ is

We begin with the linear equation $a+5b=42$, where $a$ and $b$ are natural numbers ($a,b\in \{1,2,3,\dots \}$). To find the number of solutions $m$, we isolate $a$:

Since $a\ge 1$, we apply the constraint:

Because $b$ must be a natural number, the possible values for $b$ are $\{1,2,3,4,5,6,7,8\}$. Each value of $b$ yields a unique natural number $a$, confirming that the set $R$ contains exactly $m=8$ elements.

Next, we evaluate the summation ${\sum }_{n=1}^8(1-i^{n!})$. We calculate the terms individually to identify a pattern:

For $n=1$:

For $n=2$:

For $n=3$:

We observe the behavior of $i^{n!}$ for $n\ge 4$. Since $n!$ contains the factor 4 for all $n\ge 4$, $n!$ is always a multiple of 4.

Recalling that $i^k=1$ whenever $k$ is a multiple of 4, we find that for $n\in \{4,5,6,7,8\}$:

The summation simplifies significantly:

By comparing $S=5-i$ to the form $x+iy$, we identify the real and imaginary components:

Finally, we compute the requested value $m+x+y$:

The final result is 12.