Question

pls answer me this question
FOR 25 POINTS!!

Answer 1

So our target Expression is

P + $\sqrt{p^{2}+\sqrt{p^{4}+\sqrt{p}^{8}+ \sqrt{.....}    }  }$

An indirect approach that is quite simple to consider

p(1+ $\sqrt{1+\sqrt{1+}\sqrt{....}  }$ )

It follows that we distribute the P we will have

P + P$\sqrt{1+\sqrt{....} }$

Which yields

P + $\sqrt{p^{2}+p^{2}\sqrt{1+....}   }$

ANd again we should want to distribute p4 & p8

--------------------------------------------------------------------------

But So now the trick is to evaluate:

S = 1 + $\sqrt{1+\sqrt{....} }$

It will also

$(S-1)^{2}$ = 1 + $\sqrt{1+\sqrt{....} }$

(S−1)$^{2}$=S

Using this we can solve and find

S$^{2}$−3S+1=0

Which has solutions given by the quadratic formula:

S= $\frac{3+- \sqrt{5} }{2}$

A simple test of evaluating the first few terms of

S = 1 + $\sqrt{1+\sqrt{....} }$

Suggests that solution closer to 2 is in order this we have

S= $\frac{3+- \sqrt{5} }{2}$

And this to evaluate:

P + $\sqrt{p^{2}+\sqrt{p^{4}+\sqrt{p}^{8}+ \sqrt{.....}    }  }$

Is just

$P{(\frac{3+- \sqrt{5} }{2})}$

Hope You Understand  

Thank You

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Answer 2

So our target Expression is

P +

An indirect approach that is quite simple to consider

p(1+ )

It follows that we distribute the P we will have

P + P

Which yields

P +

ANd again we should want to distribute p4 & p8

--------------------------------------------------------------------------

But So now the trick is to evaluate:

S = 1 +

It will also

= 1 +

(S−1)=S

Using this we can solve and find

S−3S+1=0

Which has solutions given by the quadratic formula:

S=

A simple test of evaluating the first few terms of

S = 1 +

Suggests that solution closer to 2 is in order this we have

S=

And this to evaluate:

P +

Is just

Hope You Understand  

Thank You