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For every positive integer n, prove …

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For every positive integer n, prove that 7 power n minus 3 power n is divisible by 4

Posted by Sujita Sujita 8 years, 3 months ago

CBSE > Class 11 > Mathematics

1 answers

Vijay Arora 8 years, 3 months ago

Step 1, For n =1 7^1-3^1 =4.which is divisible by 4. Step 2, Given statement is true for n=1, It must be true for n=k, 7^k-3^k=(M) 4 [As it is divisible by 4, let =(M) 4 where M is a positive integer.]...... (1) Step 3, For n=k+1, 7^(k+1)-3^(k+1) 7^k×7^1-3^k×3^1 Using equation (1) {(M) 4+3^k} ×7^1-3^k×3^1 (M) 4×7+3^k×7-3^k×3 7(M)4+3^k(7-3) 7(M)4+3^k×4 7(M)4+3^k×(M)4 7 is multiplied with number which is divisible by 4 and added to positive number which is also divisible by 4. Therefore the resultant is also a divisible by 4.

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