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Q.

Show that 3.52n+1+23n+1 is divisible by 17, kN.

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Detailed Solution

Let S(n)∣=3.52n+1+23n+1 is divisible by 17

put  n = 1

3.52n+1+23n+1

=3.52(1)+23(1)+1=3.52+1+23+1=3.53+24

=3(125)+16=375+16=391

=17(23) is divisible by 17

S(1) is true

Assume that S(k) is true for some kN

3.52k+1+23k+1=17m (where m is an Integer)

3.52k+1=17m23k+1     ..... (1)

to prove S(n) is true for n = k + 1

3.52(k+1)+1+23(k+1)+1

=3.52k+2+1+23k+3+1

=352k+152+23k+123

=17m23k+1(25)+23k+1(8)   (from (1))

=17m(25)1723k+1

=1725m23k+1  is divisible by 17

S(k+1) is true

By the prinicple   of finite mathematical induction
S(n) is true for all nN .

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