Question

Prove the following by using the principle of mathematical induction for all n∈N: 10^(2n – 1) + 1 is divisible by 11.

Answer 1

Given : 10 ⁽²ⁿ⁻¹⁾ + 1 is divisible by 11.

To find : to be proved by mathematical Induction

Solution:

10 ⁽²ⁿ⁻¹⁾ + 1 is divisible by 11.

n = 1

10¹ + 1 = 11 is divisible by 11.

n = 2

10³ + 1 = 1001    

1001/11 =  91

divisible by 11.

Lets assume its true for n = k

then   $10^{2k-1}$ + 1   is divisible by 11

=>   $10^{2k-1}$   = 11a  - 1

Now lets check for n  = k + 1

2(k + 1) - 1 = 2k + 1

$10^{2k+1}$ + 1

= 100($10^{2k-1}$  )  + 1

= 100(11a - 1) + 1

= 11(100a) - 100 + 1

= 11 (100a) - 99

= 11(100a - 9)

Hence divisible by 11

QED

Hence proved

10 ⁽²ⁿ⁻¹⁾ + 1 is divisible by 11.

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