Question

The value of \[\sum\limits_{n=0}^{100}{{{i}^{n!}}}\] equals (where, \[i=\sqrt{-1}\] ).

Answer 1

Hint: Expand the value of \[{{i}^{n!}}\] from n = 0 to n = 100. Thus modify them by applying their factorial value. Now substitute the values of the power of iota and simplify it.

Complete step-by-step answer:
We have been given an expression for which we should find the value.
Given to us, \[\sum\limits_{n=0}^{100}{{{i}^{n!}}}\].
First let us expand \[{{i}^{n!}}\], where n = 0, 1, 2, 3,…….., 100 as it is given from n = 0 to n = 100.
Hence we can write it as,
\[\sum\limits_{n=0}^{100}{{{i}^{n!}}}={{1}^{0!}}+{{i}^{1!}}+{{i}^{2!}}+{{i}^{3!}}+......+{{i}^{100!}}-(1)\]
We know the basics of factorial,
\[\begin{align}
  & 0!=1 \\
 & 1!=1 \\
 & 2!=2\times 1=2 \\
 & 3!=3\times 2\times 1=6 \\
 & 4!=4\times 3\times 2\times 1=24 \\
 & 5!=5\times 4\times 3\times 2\times 1=120 \\
\end{align}\]
And thus it continues like this. Hence, we can modify (1) on the above basis as,
\[\sum\limits_{n=0}^{100}{{{i}^{n!}}}={{1}^{1}}+{{i}^{2}}+{{i}^{24}}+{{i}^{120}}+......+{{i}^{100!}}-(2)\]
We know that, \[{{i}^{2}}=-1\] and we have been given, \[i=\sqrt{-1}\].
Similarly, \[{{i}^{4}}=1,{{i}^{6}}=-1\]
Thus we can substitute these values in (2).
\[\begin{align}
  & \sum\limits_{n=0}^{100}{{{i}^{n!}}}=i+i+\left( -1 \right)+\left( -1 \right)+1+1+1+......+1 \\
 & \sum\limits_{n=0}^{100}{{{i}^{n!}}}=i+i-1-1+1+1+......+1 \\
\end{align}\]
\[\sum\limits_{n=0}^{100}{{{i}^{n!}}}=2i-2+1+1+1+.....+1\], here (+1) is repeated 97 times
\[\begin{align}
  & \sum\limits_{n=0}^{100}{{{i}^{n!}}}=2i-2+97 \\
 & \sum\limits_{n=0}^{100}{{{i}^{n!}}}=2i+95 \\
\end{align}\]
Hence, we got \[\sum\limits_{n=0}^{100}{{{i}^{n!}}}=2i+95\].
Thus we got the value of the given expression as \[\left( 2i+95 \right)\].

Note: To solve this question, you should know the concept of factorial as well as complex numbers. Expand the given expression from n = 0, 1, 2,…..,100. After this it will be easy to substitute factorial value. In \[{{i}^{3!}},{{i}^{4!}},....{{i}^{100!}}\] all their values will be (+1).