© examsiri.com
Question : 3 of 160
Marks:
+1,
-0
Solution:
Let P(n)=22n+1+32n+1,n∈N
=22n⋅2+32n⋅3
=4n⋅2+9n⋅3
=2⋅(5−1)n+3⋅(10−1)n
By using bionomial expansion
(5−1)n=nC05n−nC15n−1+nC25n−2+nCn(−1)n
(10−1)n=nC010n−nC110n−1+...+nCn(−1)n
=2[nC05n−nC15n−1+nC25n−2−nC35n−3
+...+(−1)n]
+3⋅[nC010n−nC110n−1+nC210n−2−
nC310n−3+...+(−1)n]
=2⋅nC05n−2⋅nC15n−1+2⋅nC25n−2−2⋅nC35n−3
...2⋅(−1)n
+3⋅nC010n−3⋅nC110n−1+3⋅nC2102−n−3⋅nC310n−3
...+3⋅(−1)n
=nC0(2⋅5n+3⋅10n)−nC1(2⋅5n−1+3⋅10n−1)
+nC2(2⋅5n−2+3⋅10n−2)
−nC3(2⋅5n−3+3⋅10n−3)+...+(−1)n5
=5[nC0(2⋅5n−1+3⋅2n⋅5n−1)
−nC1(2⋅5n−2+3⋅2n−1⋅5n−2)
+nC2(2⋅5n−3+3⋅2n−2⋅5n−3)
P(n)=5K=−C3(2⋅5n−4+3⋅2n−3⋅5n−4)+...+(−1)n
P(n)=5K
where, K=nC0(2⋅5n−1+3⋅2n5n−1)
−nC1(2⋅5n−2+3⋅2n−1⋅5n−2)+...+(−1)n]
∴22n+1+33n+1 is divisible by 5,∀n∈N.
© examsiri.com
Go to Question: