# Prove that N3 - 7n + 3 is Divisible by 3 for All N ∈ N . - Mathematics

Prove that n3 - 7+ 3 is divisible by 3 for all n $\in$ N .

#### Solution

$\text{ Let } p\left( n \right) = n^3 - 7n + 3 \text{ is divisible by } 3 \forall n \in N .$

$\text{ Step I: For } n = 1,$

$p\left( 1 \right) = 1^3 - 7 \times 1 + 3 = 1 - 7 + 3 = - 3, \text{ which is clearly divisible by } 3$

$\text{ So, it is true for n } = 1$

$\text{ Step II: For } n = k,$

$\text{ Let } p\left( k \right) = k^3 - 7k + 3 = 3m, \text{ where m is any integer, be true } \forall k \in N .$

$\text{ Step III: For } n = k + 1,$

$p\left( k + 1 \right) = \left( k + 1 \right)^3 - 7\left( k + 1 \right) + 3$

$= k^3 + 3 k^2 + 3k + 1 - 7k - 7 + 3$

$= k^3 + 3 k^2 - 4k - 3$

$= k^3 - 7k + 3 + 3 k^2 + 3k - 6$

$= 3m + 3\left( k^2 + k + 2 \right) \left[ \text{ Using step } II \right]$

$= 3\left( m + k^2 + k + 2 \right)$

$= 3p, \text{ where p is any integer }$

$\text{ So,} p\left( k + 1 \right) \text{ is divisible by } 3 .$

Hence, n3  - 7+ 3 is divisible by 3 for all n $\in$ N .

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 12 Mathematical Induction
Exercise 12.2 | Q 29 | Page 28