Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 10 - Section 10.4 - Mathematical Induction - Exercise Set - Page 1085: 36


The value at $ n=41$ of this mathematical expression, “ ${{n}^{2}}-n+41$ ” is not a prime number.

Work Step by Step

Let us consider the expression: ${{S}_{n}}$: ${{n}^{2}}-n+41$ For $ n=41$, $\begin{align} & {{S}_{n}}={{n}^{2}}-n+41 \\ & {{S}_{n}}=n(n-1)+41 \\ & {{S}_{n}}=(41\times 40)+41 \\ & {{S}_{41}}=(41)(40+1) \\ & {{S}_{41}}={{41}^{2}} \\ \end{align}$ Since, the ${{S}_{41}}$ is a perfect square; hence it is not a prime number Thus, in the figure, one can see that the domino is true until $ n=40$ but after that it does not follow; the second condition of mathematical induction does not follow and hence the domino at $ n=41$ does not fall, thereby breaking the sequence.
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