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Question: If \({a_n} = {i^{{{(n + 1)}^2}}}\) , where \(i = \sqrt { - 1} \) and n=1,2,3 ... n, then the value o...

If an=i(n+1)2{a_n} = {i^{{{(n + 1)}^2}}} , where i=−1i = \sqrt { - 1} and n=1,2,3 ... n, then the value of a1+a3+a5+ ... +a25{a_1} + {a_3} + {a_5} + \,...\, + {a_{25}} is
A) 1313
B) 13−i13 - i
C) 13+i13 + i
D) 1212
D) 12−i12 - i

Explanation

Solution

So, they have asked to find the sum of series of a1+a3+a5+ ... +a25{a_1} + {a_3} + {a_5} + \,...\, + {a_{25}} and they have given an=i(n+1)2{a_n} = {i^{{{(n + 1)}^2}}}. Now, to find the solution for the given problem, first you need to see how many terms are present in it. Then put their term values in an=i(n+1)2{a_n} = {i^{{{(n + 1)}^2}}} this equation and remember that they also given one more thing and that it i=−1i = \sqrt { - 1} and after remembering this you just need to put value and calculate it to get correct answer.

Complete step by step answer:
First of all they given,
⇒an=i(n+1)2\Rightarrow {a_n} = {i^{{{(n + 1)}^2}}}
Now, we have to look at the given series. Given series is,
⇒a1+a3+a5+ ... +a25\Rightarrow {a_1} + {a_3} + {a_5} + \,...\, + {a_{25}}
If you look at the series you will figure out that only 1313 terms are present in it. Now, count this series as 1313 odd number as we can see. So, we can put values of it in an=i(n+1)2{a_n} = {i^{{{(n + 1)}^2}}} this equation and we will get following terms,
⇒a1+a3+a5+ ... +a25=i(1+1)2+i(3+1)2+i(5+1)2+ ... +i(25+1)2\Rightarrow {a_1} + {a_3} + {a_5} + \,...\, + {a_{25}} = {i^{{{(1 + 1)}^2}}} + {i^{{{(3 + 1)}^2}}} + {i^{{{(5 + 1)}^2}}} + \,...\, + {i^{{{(25 + 1)}^2}}}
From further simplification,
⇒a1+a3+a5+ ... +a25=i(4)+i(16)+i(36)+ ... +i(676)\Rightarrow {a_1} + {a_3} + {a_5} + \,...\, + {a_{25}} = {i^{(4)}} + {i^{(16)}} + {i^{(36)}} + \,...\, + {i^{(676)}}
Now, as we can have given that i=−1i = \sqrt { - 1} so we can clearly say that i4=1{i^4} = 1
So as there are 1313 terms so we can rewrite above equation as bellow,
⇒a1+a3+a5+ ... +a25=1+1+1+ ... +1\Rightarrow {a_1} + {a_3} + {a_5} + \,...\, + {a_{25}} = 1 + 1 + 1 + \,...\, + 1
So there are 1313 terms so the sum of the given series is 1313 .
Therefore, There are 13 terms in the series. Hence, option (A) is the correct option.

Note:
In this question they have given i=−1i = \sqrt { - 1} so this is nothing but it is a complex number. It contains imaginary numbers and ii is called “iota”. Basically the complex number is the combination of a real number and an imaginary number. Hence, a complex number is a simple representation of the addition of two numbers, i.e., real number and an imaginary number.