Answer:
If one amoeba dies then two amoeba get born:
at different stages of the generations of the amoeba it changes accordingly and the died amoeba numbers are also change in the given manner :
stages dies born
I 1 2
II [tex]2^{1}[/tex] 4 = [tex]2^{2}[/tex]
III [tex]2^{2}[/tex] 8 = [tex]2^{3}[/tex]
At [tex]n^{th}[/tex] = [tex]2^{n}[/tex] [tex]2^{n+1}[/tex]
By the geometric series formula:
1+[tex]a^{1}[/tex]+ [tex]a^{2}[/tex]+ [tex]a^{3}[/tex]...+ [tex]a^{n}[/tex]=[tex]\frac{a^{n+1} }{a^-1} }[/tex]
so, 1+[tex]2^{1}[/tex]+ [tex]2^{2}[/tex]+ [tex]2^{3}[/tex]...+ [tex]2^{n}[/tex]=[tex]\frac{2^{n+1} }{2^-1} }[/tex]
Thus, total amoeba died = [tex]2^{n+1}[/tex]-1
total amoeba died+1 = [tex]2^{n+1}[/tex] (born).... hence prooved
