Exercise · ZW-MATHEMATICAL-INDUCTION-006 · ★★☆ Intermediate

Using \(1^2+2^2+\cdots+n^2=\tfrac{n(n+1)(2n+1)}{6}\), compute \(1^2+2^2+\cdots+6^2\).

Choose one option.

Show answer

C. \(91\)

Explanation

With \(n=6\): \(\tfrac{6\cdot7\cdot13}{6}=91\). The value \(441=21^2\) comes from squaring the sum \(1+\cdots+6\).

Review the concept: Mathematical induction

Mathematical induction proves that a property \(P(n)\) holds for every integer \(n\ge n_0\): one checks the base case \(P(n_0)\) and proves that, for each \(k\ge n_0\), \(P(k)\) implies \(P(k+1)\).

\[\big[P(n_0)\wedge\forall k\ge n_0\,(P(k)\Rightarrow P(k+1))\big]\Rightarrow\forall n\ge n_0\,P(n)\]

More practice problems: mathematical induction

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ZW-MATHEMATICAL-INDUCTION-005 In the inductive step of 1+2+cdots+n=tfracn(n+1)2, adding k+1 to both sides of P(k) makes the right…
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ZW-MATHEMATICAL-INDUCTION-009 Let P(n): «n=n+1». It is proved that P(k)⇒ P(k+1) by adding 1 to both sides. What follows?