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Question

Show that 24n+415n16, where nN is divisible by 225.

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Solution

We have,

24n+415n16=24(n+1)15n16

=16n+115n16=(1+15)n+115n16

=n+1C0150+n+1C1151+n+1C2152+n+1C3153+....+n+1Cn+1(15)n+115n16

=1+(n+1)15+n+1C2152+n+1C3153+....+n+1Cn+1(15)n+115n16

=1+15n+15+n+1C2152+n+1C3153+...+n+1Cn+1(15)n+115n16

=152[n+1C2+n+1C315+....so on]

Thus, 24n+415n16 is divisible by 225.


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