Equality versus estimate In an identity proof, algebra aims at exact equality. In an inequality proof, the inductive step must preserve the correct direction while building an estimate strong enough to reach the next target. Inequality workflow Verify the stated starting value; inequality claims often begin at \(n=2\), \(3\) or later. Write the target inequality for \(k+1\). Use the induction hypothesis in the correct direction. Prove the remaining comparison from the domain condition \(k\ge n_0\). ObstacleUseful responseExample of the response A multiplicative left sideCompare growth factors.Show the new factor on the left exceeds the factor needed on the right. A polynomial targetBound \(k+1\) by a multiple of \(k\).From \(k\ge4\), use \(k+1\le\frac54k\). The hypothesis is too weakStrengthen the claimed estimate and prove the stronger statement.Replace a barely sufficient bound by one that survives the next step. The inequality direction changesCheck the sign before multiplying or dividing.A negative multiplier reverses the inequality. Sign check From \(A_k>B_k\), multiplying by a positive number preserves the direction. Multiplying by a negative number reverses it. Every multiplier or divisor used in an inductive inequality must have a known sign. Starting values carry meaning If the base case fails, do not conceal it. Determine whether the statement is false or whether its correct domain starts later. Class check To prove \(3^n>n^3\) for \(n\ge4\), explain why comparing \(3k^3\) with \((k+1)^3\) is the remaining task after the induction hypothesis is used.

Induction for Inequalities

IB Year 5 | Grade 11

Equality versus estimate

In an identity proof, algebra aims at exact equality. In an inequality proof, the inductive step must preserve the correct direction while building an estimate strong enough to reach the next target.

Inequality workflow

Verify the stated starting value; inequality claims often begin at \(n=2\), \(3\) or later.
Write the target inequality for \(k+1\).
Use the induction hypothesis in the correct direction.
Prove the remaining comparison from the domain condition \(k\ge n_0\).

Obstacle	Useful response	Example of the response
A multiplicative left side	Compare growth factors.	Show the new factor on the left exceeds the factor needed on the right.
A polynomial target	Bound \(k+1\) by a multiple of \(k\).	From \(k\ge4\), use \(k+1\le\frac54k\).
The hypothesis is too weak	Strengthen the claimed estimate and prove the stronger statement.	Replace a barely sufficient bound by one that survives the next step.
The inequality direction changes	Check the sign before multiplying or dividing.	A negative multiplier reverses the inequality.

Sign check

From \(A_k>B_k\), multiplying by a positive number preserves the direction. Multiplying by a negative number reverses it. Every multiplier or divisor used in an inductive inequality must have a known sign.

Starting values carry meaning

If the base case fails, do not conceal it. Determine whether the statement is false or whether its correct domain starts later.

Class check

To prove \(3^n>n^3\) for \(n\ge4\), explain why comparing \(3k^3\) with \((k+1)^3\) is the remaining task after the induction hypothesis is used.

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