Question

the least positive value of x satisfying 2 power (3+3cosx+3cos2x+3cos3x+....)= 2power6 will be where |cosx | <1)

Answer 1

Appropriate Question :-

The least positive value of x satisfying

$\rm \: {2}^{3 + 3cosx + {3cos}^{2}x + {3cos}^{3} x + \cdots} \: = \: {2}^{6} \: such \: that \: |cosx| < 1 \: is \cdots\cdots$

$\color{green}\large\underline{\sf{Solution-}}$

Given expression is

$\rm \: {2}^{3 + 3cosx + {3cos}^{2}x + {3cos}^{3} x + \cdots} \: = \: {2}^{6} \:$

Now, Consider

$\rm \: 3 + 3cosx + {3cos}^{2}x + {3cos}^{3}x + \cdots$

Its an infinite GP series with

• First term, a = 3

• Common ratio, r = cosx

We know,

Sum of infinite GP series with first term a, common ratio r is given by

$\color{green}\boxed{ \rm{ \: \: S_ \infty \: = \: \frac{a}{1 \: - \: r} \: \: \: \: provided \: that \: |r| < 1 \: }}$

So,

$\rm \: 3 + 3cosx + {3cos}^{2}x + {3cos}^{3}x + \cdots = \frac{3}{1 -cosx}$

Now, given expression

$\rm \: {2}^{3 + 3cosx + {3cos}^{2}x + {3cos}^{3} x + \cdots} \: = \: {2}^{6} \:$

can be rewritten as

$\rm \: 3 + 3cosx + {3cos}^{2}x + {3cos}^{3}x + \cdots\cdots = 6$

can be further rewritten as

$\rm \: \dfrac{3}{1 - cosx} = 6$

$\rm \: 1 - cosx = \dfrac{3}{6}$

$\rm \: 1 - cosx = \dfrac{1}{2}$

$\rm \: - cosx = \dfrac{1}{2} - 1$

$\rm \: - cosx = \dfrac{1 - 2}{2}$

$\rm \: - cosx = \dfrac{ - 1}{2}$

$\rm \: cosx = \dfrac{ 1}{2}$

$\rm \: cosx = cos\dfrac{\pi}{3}$

$\rm\implies \:x =\dfrac{\pi}{3}$

Hence, Least positive value of x satisfying the equation

$\color{green}\boxed{\rm{{2}^{3 + 3cosx + {3cos}^{2}x + {3cos}^{3} x + \cdots} = {2}^{6} \: such \: that \: |cosx| < 1 \: is \dfrac{\pi}{3}}}$