written 7.7 years ago by | modified 2.6 years ago by |
Mumbai University > Computer Engineering > Sem 3 > Discrete Structures
Marks: 5 Marks
Year: Dec 2015
written 7.7 years ago by | modified 2.6 years ago by |
Mumbai University > Computer Engineering > Sem 3 > Discrete Structures
Marks: 5 Marks
Year: Dec 2015
written 7.7 years ago by |
Step1: Basis of induction
For n=1
We have
$6^{n+2} + 7^{2n+1} = 6^3 + 7^3 =559$ is divisible by 43
Step2: Induction step: Assume that $6^{k+2} - 7^{2k+1}$ is divisible by 43. Then we have
$6^{k+3} - 7^{2k+2}$ $=6^{k+2}. 6 -7^{2k+2} \\ =6^{k+2}.(14-7) -7^{2k+2} \\ =14(6^{k+2})-7(6^{k+2})- 7^{2k+2} \\ =14(6^{k+2})-7(6^{k+2}+ 7^{2k+1})$
Since both forms in this sum are divisible by 43(the first because it is 43 times an integer and the second by the assumption of the induction step), it follows that $6^{k+2} - 7^{2k+1}$ is also divisible of 43. Thus, by the principle of mathematical induction, $6^{n+2} — 7^{2n+1}$ is a divisible of 43.