We need `1 + 2 + 4 + 8 + ... + 2^63`

Now `a_1= 1`, `r = 2`, `n = 64`.

Our formula for the sum to n terms says:

`S_n=(a_1(1-r^n))/(1-r)\ (r!=1)`

Substituting our values:

`S_64=(1(1-2^64))/(1-2)`

`=1.84467xx10^19\ "grains"`

Each grain weighs `20\ "mg" = 2 × 10^-5\ "kg" ` `= 2 × 10^-8\ "tonnes"`.

So the weight is

`(1.84467 × 10^19) × ` `(2 × 10^-8)\ "tonnes" ` `= 369\ "billion tonnes"`, so of course, the king cannot grant the Prince's wish.

NOTE 1: There are `1000\ "kg"` in one tonne.

NOTE 2: The world annual output of rice today is only `600` million (not billion) tonnes!

NOTE 3: We are using the US/French 'billion' (`10^9`) and not the British 'billion' (`10^12`). [See Short and Long Scales.]

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