Question

6. Use a proof by induction to show that the sum 1 +5 + 25 + 125 + ... +5n-1 = 51-1 (6 marks) 4 7. You are visiting another country. They do not use coins, so all prices are in whole numbers (no fractions or decimals). In addition, they are suspicious of even numbers, and so have printed only $3 and $5 bills. And since they have never taken discrete math, they are not very good at making change. Use a proof by STRONG induction to show that for every item priced $8 and above you can make exact change using only $3 and $5 bills. (6 marks)

please write each step and explain so I can understand, not judt have the answers

Answer 1

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To prove, 1+ 5 + 25+125+.. (0-1) = -1 4 for nal. Step 1: At first we check Substitu tuting the value n=1 in the equation get, 5(1-1) we 5- 4 5 5- !! 4 4 [since 5 - 1 1 = We can say it is true fer mal

16 17 18 19 20 21 22 23 24 25 26 27 28 29 31-335 / Week 5 Step 2 Now we assume that it is true for Nak. Substituting the value in nation we get, 1+5+ 25 + 125t..+5 K- the equation k ~ +5k 5 4 . It is true for nek. Now we assume na kt! (k+1-1) +5 1+5 +25+125t.. +5 From eq lil we can say, K = 54 - 1 +51k tay 4 K 5k 1 + 3 2020

22 23 24 25 26 27 28 29 30 31 34-332 / Week 6 5'-) +4.5 taling sk commen, 4. ski 4+)-1 4 5%(5) - 4 = 5K+ - 4 LHS=RHS We can say this by using Mathematical induction. It is true for all n EN