COMEDK2023MathematicsMathematical Induction

2^ 3 n -7 n-1 is divisible by

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Correct answer

C. 49

Step-by-step solution

aligned & Let P(n)=2^ 3 n -7 n-1 & P(1)=2^ 3(1) -7(1)-1=8-8=0 & P(2)=2^ 3(2) -7(2)-1=64-15=49 & aligned P(1) and P(2) are divisible by 49 . Let P(k)=2^ 3 k -7 k-1=49 t , where t is an integer Now, aligned P(k+1) & =2^ 3(k+1) -7(k+1)-1=2^ 3 k 2^3-7 k-7-1 & =8 (2^ 3 k -7 k-1 )+49 k & =8(49 t)+49 k & =49(8 t+k), where 8 t+k is an integer aligned Thus, 2^ 3 n -7 n-1 is divisible by 49 .

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