The Martian monetary system uses colored beads instead of coins. A blue bead is worth 3 Martian credits, and a red bead is worth 7 Martian credits. Thus, three blue beads are worth 9 credits, and a blue and red bead together are worth 10 credits, but no combination of blue and red beads is worth 11 credits. Using Induction, prove that for all n ≥ 12, there is some combination of blue and red beads that is worth n credits. Clearly state the type of Induction you have used to prove the problem

"a" is showing the number of blue beads

"b" represents the number of red beads in a combination

let "n" show the number of credits.

We get the following equation:

n = 3a + 7b.

Now, we want to prove that for all n ≥ 12, there is some combination of blue and red beads that is worth n credits, there are a, b in N such that:

n = 3a + 7b

Hence we will prove this by induction.

Base Cases: n = 12.

Applying the formula clearly 12 = 3*4+7*0.

n = 13

Applying the formula clearly 13 = 3*2+7*1.

n = 14.

Applying the formula clearly 14 = 3*0+7*2.

Induction Step: Assume that for all 12<=k<n there are x, y in N such that

k = 3x + 7y. We will now prove that there are a, b in N such that n = 3a + 7b.

By the induction hypothesis we know that there are x, y in N such that n-3 = 3x + 7y.

Hence n = n - 3 + 3 = 3x + 7y + 3 = 3(x + 1) + 7y, so choosing a = x + 1

and b = y we see that n = 3a + 7b.

Therefore by induction, the claim is true for all n ≥ 12

not enough information

step-by-step explanation:

we cannot know this, because the other students could have handed out 10 flyers, or they could have handed out 100. one way, the mean would go up, and another way, the mean would go down.