Question

How do you find the sum of the series \[{i^2}\] from $i = 1$ to 12.

Answer 1

Hint: According to the question given in the question we have to determine the sum of the series \[{i^2}\] from $i = 1$ to 12. So, first of all to find the sum we have to determine the values of \[{i^2}\]and \[{i^3}\] which are explained as below:

Formula used:
$ \Rightarrow {i^2} = - 1..............(A)$and,
$ \Rightarrow {i^3} = - i..............(B)$
Hence, on substituting the formula in the series we can determine the sum of the series but before that we have to determine the series which can be done as mentioned in the question that we have to determine the sum form $i = 1$ to 12.
Now, after substituting all the values in the series having terms in the form of \[{i^2}\], \[{i^3}\] and \[{i^{12}}\] but before that we have to convert all the terms in the form of \[{i^2}\] and \[{i^3}\] so, that we can easily substitute the values in the series.

Complete step-by-step answer:
Step 1: First of all we have to write the series for \[{i^2}\] as mentioned in the question that we have to find the sum from $i = 1$ to 12. Hence,
$ = {i^1} + {i^2} + {i^3} + {i^4} + {i^5} + {i^6} + {i^7} + {i^8} + {i^9} + {i^{10}} + {i^{11}} + {i^{12}}................(1)$
Step 2: Now, to obtain the value of the series we have to apply the formulas (A) and (B) as mentioned in the solution hint in the expression (1) which is as obtained in the solution step 1. Hence,
$ = i - 1 - i + {i^4} + {i^5} + {i^6} + {i^7} + {i^8} + {i^9} + {i^{10}} + {i^{11}} + {i^{12}}................(2)$
Step 3: Now, to find the sum of the series (2) which is as obtained in the solution step 2 we have to convert all the terms in form of \[{i^2}\] and \[{i^3}\] which can be as below:
$ = i - 1 - i + {i^4} + {i^4} \times i + {({i^2})^3} + {({i^2})^3}i + {({i^4})^2} + {i^8} \times i + {({i^2})^5} + {({i^2})^5}i + {({i^4})^3}............(3)$
Ste 4: Now, we just have to substitute all the values in the series (3) which is as obtained in the solution step 3. Hence,
$ = i - 1 - i + 1 + 1 \times i + {( - 1)^3} + {( - 1)^3}i + {(1)^2} + 1 \times i + {( - 1)^5} + {( - 1)^5}i + {(1)^3}$………….(4)
 Step 5: Now, we just have to solve the series (4) which is as obtained in the solution step 4 by adding and subtracting all the terms of the series. Hence,
$
   = i - 1 - i + 1 + i - 1 - i + 1 + i - 1 - i + 1 \\
   = 0 \\
 $

Final solution: Hence, with the help of the formula (A), (B) we have determined the sum of the series which is 0.

Note:
To obtain the value of ${i^4},{i^5}$ and all the terms of the series after these term till ${i^{12}}$ we have to convert the terms in the form of \[{i^2}\] and \[{i^3}\] so, that we can easily substitute the values.
After substituting all the values in the series having terms in the form of\[{i^2}\], \[{i^3}\] and \[{i^{12}}\] but before that we have to convert all the terms in the form of \[{i^2}\] and \[{i^3}\] so, that we can easily substitute the values in the series