p( n)=2.7n+3.5n-5is divisible by 24
Answers
Step-by-step explanation:
Let, p(n) denotes the statement that 2.7ⁿ+3.5ⁿ-5 is divisible by 24.
For n=1, p(1): 2.7+3.5-5=14+15-5=29-5=24 which is divisible by 24.
Let us assume that p(n) is true for n=k i.e.,
2. 7^{k}+3. 5^{k}-52.7
k
+3.5
k
−5 is divisible by 24.
Then for n=k+1 p(k+1): 2. 7^k+1}+3. 5^{k+1}-5
∴, 2{p(k+1)-p(k)}
=2{(2.7^{k+1}+3. 5^{k+1} -52.7
k+1
+3.5
k+1
−5 )-(2. 7^{k}+3. 5^{k}-52.7
k
+3.5
k
−5 )}
=2(2. 7^{k}.7+3. 5^{k}.5-5-2. 7^{k}-3. 5^{k}+52.7
k
.7+3.5
k
.5−5−2.7
k
−3.5
k
+5 )
=2(14. 7^{k}+15. 5^{k})-4. 7^{k}-6. 5^{k}2(14.7
k
+15.5
k
)−4.7
k
−6.5
k
=28. 7^{k}+30. 5^{k}- 4. 7^{k}-6. 5^{k}28.7
k
+30.5
k
−4.7
k
−6.5
k
=24. 7^{k}+24. 5^{k}24.7
k
+24.5
k
=24( 7^{k}+ 5^{k})24(7
k
+5
k
)
which is divisible by 24
∴, p(k+1) is divisible by 24 since p(k) is divisible by 24.
Now, p(1) is true and p(k+1) is true if we assume that p(k) is true.
∴, by the principle of mathematical induction, 2.7ⁿ+3.5ⁿ-5 is divisible by 24 for all n belongs to N.