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Standard XII

Mathematics

Sequence

Let a sequenc...

Question

Let a sequence $x_{1}, x_{2}, x_{3}, . . .$ of complex number be defined by $x_{1} = 0, x_{n + 1} = x_{n}^{2} - i$ for $n > 1$ where $i^{2} = - 1$ . Find the distance of $x_{2000}$ from $x_{1997}$ in the complex plane. If the distance is $k$ , then find $k^{2}$ .

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Solution

$x_{1} = 0$
$x_{2} = 0^{2} - i = - i$
$x_{3} = {(- i)}^{2} - i = - 1 - i = - (1 + i)$
$x_{4} = {[- (1 + i)]}^{2} - i = 2 i - i = i$
$x_{5} = {(i)}^{2} - i = - 1 - i = x_{3}$
$∴$ $x_{6} = x_{4}$ and hence $x_{7} = x_{5}$ and so on,
$∴$ $x_{2 n} = i$ , $x_{2 n + 1} = - (1 + i)$
$∴$ $x_{2000} = i = (0, 1)$ and $x_{1997} = - 1 - i = (- 1, - 1)$ in the complex plane.
So, the distance between $x_{2000}$ and $x_{1997}$ is $\sqrt{1 + 4} = \sqrt{5}$ .
$∴ k^{2} = 5$

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[NCERT EXEMPLAR]

Q. Let

x_{1}, x_{2}, x_{3}, . . ., x_{n}

be a sequence of integers such that:
i) -1

\leq

x_{i}

\leq

2 for i = 1, 2 ..., n
ii)

x_{1} + x_{2} + x_{3} + . . . + x_{n}

= 19
iii)

x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + . . . + x_{n}^{2}

= 99
Determine the minimum and maximum possible values of

x_{1}^{3} + x_{2}^{3} + x_{3}^{3} + . . . + x_{n}^{3}

Q. Let

x_{1}, x_{2}, x_{3}, . . ., x_{n}

be a sequence of integers such that:
i) -1

\leq

x_{i}

\leq 2

for i = 1, 2 ..., n
ii)

x_{1}, x_{2}, x_{3}, . . ., x_{n}

= 19
iii)

x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, . . ., x_{n}^{2}

= 99
Determine the minimum and maximum possible values of

x_{1}^{3}, x_{2}^{3}, x_{3}^{3}, . . ., x_{n}^{3}

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Let a sequence x1,x2,x3,... of complex number be defined by x1=0,xn+1=x2n−i for n>1 where i2=−1. Find the distance of x2000 from x1997 in the complex plane. If the distance is k, then find k2.

x1=0x2=02−i=−ix3=(−i)2−i=−1−i=−(1+i)x4=[−(1+i)]2−i=2i−i=ix5=(i)2−i=−1−i=x3∴ x6=x4 and hence x7=x5 and so on,∴ x2n=i, x2n+1=−(1+i)∴ x2000=i=(0,1) and x1997=−1−i=(−1,−1) in the complex plane.So, the distance between x2000 and x1997 is √1+4=√5.∴k2=5

Let a sequence $x_{1}, x_{2}, x_{3}, . . .$ of complex number be defined by $x_{1} = 0, x_{n + 1} = x_{n}^{2} - i$ for $n > 1$ where $i^{2} = - 1$ . Find the distance of $x_{2000}$ from $x_{1997}$ in the complex plane. If the distance is $k$ , then find $k^{2}$ .