Let S be the set of positive integers that can be written as a sum of one or mor

Question

Let S be the set of positive integers that can be written as a sum of one or more 4's and/or 7's. For example, 7 e S (in set) and 18 e S (because 18 = 4+7+7).

It turns out that S contains all positive integers except for some relatively small missing values.

(a) What is the greatest integer g that is not in S? Just write down the answer, you dont have to show your work here or prove that g is not in S.

(b) Prove using mathematica induction that all integers greater than g are elements of S. (HINT: there are four base cases that need to be checked.)

Explanation / Answer

a.17

b.

we can have18(4+7+7)

we can have 19(4+4+4+7)

we can have 20(4+4+4+4+4)

we can have 21(7+7+4)

as we got 4 consecutive numbers we can get next 4 numbers by adding 4 to these numbers and so on .we will have all numbers >17 in S

for n>17 we can have all the numbers in S

if we want 22 we can add 4 for 18 and similarly if we want 23 we can add 4 for 19 .if we want 24 we can add 4 for 20...and if we want 25 we can add 4 to 21 and if we want 26 add 4 to 22 and so on....we can get all number this way.

let n be in S and if n=18+x (x be some number )

then n+1 is also in S and n+1 = 19+x

then n+2 is also in S and n+2 = 20+x

then n+3 is also in S and n+3 = 21+x

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