Question

An amoeba reproduces by splitting itself into two amoebas. This means that the original amoeba dies and its two offspring live on. Prove that if you start off with one amoeba then after any amount of time, the number of living amoebas is equal to one plus the number of amoebas who have died.

Answer 1

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
`2^(1)` 4 =
`2^(2)`

III
`2^(2)` 8 =
`2^(3)`

At
`n^(th)` =
`2^(n)`
`2^(n+1)`

By the geometric series formula:

1+
`a^(1)`+
`a^(2)`+
`a^(3)`...+
`a^(n)`=
`(a^(n+1) )/(a^-1) }`

so, 1+
`2^(1)`+
`2^(2)`+
`2^(3)`...+
`2^(n)`=
`(2^(n+1) )/(2^-1) }`

Thus, total amoeba died =
`2^(n+1)`-1

total amoeba died+1 =
`2^(n+1)` (born).... hence prooved