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Answer
Correct option: C.$1$ જો $n$ અયુગ્મ હોય, $-1$ જો $n$ યુગ્મ હોય
c
(c)Let $z = {i^{[1 + 3 + 5 + .... + (2n + 1)]}}$
Clearly series is A.P. with common difference $= 2$
$\because \,{T_n} = 2n - 1$and ${T_{n + 1}} = 2n + 1$
So, number of terms in A. P. $ = n + 1$
Now, ${S_{n + 1}} = \frac{{n + 1}}{2}[2.1 + (n + 1 - 1)2]$
$ \Rightarrow {S_{n + 1}} = \frac{{n + 1}}{2}[2 + 2n] = (n + 1)^2$ i.e. $i^{(n + 1)^2}$
Now put $n = 1,\,2,\,3,\,4,\,5,\,.....$
$n = 1,z = {i^4} = 1$, $n = 2,\,z = {i^6} = - 1$,
$n = 3,\,z = {i^8} = 1$, $n = 4,\,z = {i^{10}} = - 1$,
$n = 5,\,\,z = {i^{12}} = 1\,,........$
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