Question

If a+b=1, prove by induction that a^3+b^3 is bigger or equal to ¼.

Answer 1

To prove that a^3+b^3 is bigger or equal to ¼, we can use mathematical induction. This involves two steps:

1. Show that the statement is true for the base case of a and b (i.e., when a and b are equal to some specific values).
2. Assume that the statement is true for some arbitrary values of a and b (i.e., the "inductive hypothesis"). Then, show that it must also be true for the next values of a and b (i.e., a+1 and b+1).

To begin, let's consider the base case where a and b are equal to 0. In this case, a^3+b^3 is equal to 0^3+0^3, which is 0. Since 0 is greater than or equal to ¼, the statement is true for the base case.

Next, we can assume that the statement is true for some arbitrary values of a and b (i.e., the inductive hypothesis). That is, we can assume that a^3+b^3 is bigger or equal to ¼.

To complete the proof, we need to show that a^3+(b+1)^3 is also bigger or equal to ¼. To do this, we can use the fact that (a+1)^3 = a^3+3a^2+3a+1. Since the inductive hypothesis states that a^3+b^3 is bigger or equal to ¼, we can say that a^3+(b+1)^3 = a^3+b^3+3a^2+3a+1 is also bigger or equal to ¼.

Therefore, by mathematical induction, we have shown that a^3+b^3 is bigger or equal to ¼ for all values of a and b.