Numbers & Place Value

Depth · one concept from first principles · step 2 of 4

Chain the given inequality twice

We are given a fact: for nonnegative m, n, 2^m+2ⁿ ≤ 1+2^(m+n), with equality when m=0 or n=0. For nonnegative a, b, c with a+b+c=4, find the largest value of 2^a+2^b+2^c.
Expressions
Your answer
How to solve
Strategy Identify Subproblems — The given fact handles two powers at a time. Use it on a and b, then treat a+b as a single exponent and use it again with c. That leaves only 2^(a+b+c)=2⁴.
1STEP 1

Use the fact on a and b

a and b are nonnegative, so the fact applies with m=a, n=b.

2^a+2^b ≤ 1+2^(a+b)
2STEP 2

Add 2^c to both sides

Adding the same number to both sides keeps the inequality.

2^a+2^b+2^c ≤ 1+2^(a+b)+2^c
3STEP 3

Use the fact on a+b and c

a+b is also nonnegative, so apply the fact with m=a+b, n=c.

2^(a+b)+2^c ≤ 1+2^(a+b+c)
4STEP 4

Chain the inequalities

Put the two steps together and use a+b+c=4.

2^a+2^b+2^c ≤ 1+1+2^(a+b+c)=2+2⁴=18
5STEP 5

Check the bound is reached

Equality needs a=0 or b=0, and a+b=0 or c=0. Try a=0, b=0, c=4.

2⁰+2⁰+2⁴=1+1+16=18
Answer
18
Other splits give less: (2,2,0) gives 4+4+1=9, (3,1,0) gives 8+2+1=11, and (4/3,4/3,4/3) gives about 7.6. Putting all of the 4 into one exponent makes one term large, which makes sense for doubling.
Takeaway

A fact about two numbers can handle three -- glue two of them into one block and use it twice.

  • Use the fact on a and b
  • Add 2^c to both sides
  • Use the fact on a+b and c
  • Chain the inequalities
  • Check the bound is reached
Where next?
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
Rewrite in base 2 and factor the sumPair 2s with 5s to make powers of ten

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▶ Practice — 12 problems