Question

Prove by mathematical induction that 7x16n-1 +3. divisible by 5 for all n belongs N.​

Answer 1

Answer:

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Step-by-step explanation:

10  

n

+3.4  

n+2

+5

Put n=1

10+3(64)+5=207 which is divisible by 3,9,23,207

We will check for n=2

100+768+5=873  

which is divisible by 3 and 9 only.

Every number divisible by 9 is also divisible by 3.

Now, let P(n) is true  

i.e. 10  

n

+3.4  

n+2

+5 is divisible by 9

⇒10  

n

+3.4  

n+2

+5=9λ       ....(1)

Now,  we will check for P(n+1)

Consider, 10  

n+1

+3.4  

n+3

+5

=10  

n

10+3.4  

n+2

4+5

=10(9λ−3.4  

n+2

−5)+3.4  

n+2

4+5    (by (1))

=90λ−27.4  

n+2

−45

=9(10λ−3.4  

n+2

−5)

which is divisible by 9.

Hence, by mathematical induction given expression is divisible by 3,9.

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