Question

114. If (-√3-i)^30 = -4^k, then the value of k is
(B) 20​

Answer 1

Answer:

20

Step-by-step explanation:

Answer 2

If (-√3-i)^30 = -4^k the value of k is 2k -3

Given:

(-√3-i)^30 = -$4^{k}$

To Find:

The value of k from the equation

Solution:

First, we take the summation n ends from 2 to 29

∑ n=2 to 29 $(1.5)^{n}$ = S

Taking the summation which tends from 1 to 30

∑ n =1 to 30 $(1.5)^{n}$ = 1.5+$1.5^{2}$ +$1.5^{3}$ +...+$1.5^{29}$+$1.5^{30}$

The given series is in Geometric progression,

Since, these are in GP,

We have the formula for Geometric progression,

S = $\frac{a(r^n -1 )}{r-1}$

Upon substituting the values in the place of r, a and n

We get,

= 1.5 ($1.5^{30 }-1$)/1.5-1

= 3($1.5^{30}$ −1)

Taking $1.5^(30)$ as k

Replace the number with K,

We get,

=3(k−1)

=1.5+$1.5^{2}$ +$1.5^{3}$ +...+$1.5^{29}$ +$1.5^{30}$

Here, summation n tends from 2 to 29 as the first summation in the

first equation as seen,

We get the value,

= 1.5+∑ n =2 to 29 $(1.5)^{n}$ + $1.5^{30}$

= 2k-4.5

= 2k -3

Hence, the value of k is 2k-3

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