Question

Prove that 2.7 n +3.5 n -5 is divisible by 24,for all n belongs to N.

Answer 1

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., 
$2. 7^{k}+3. 5^{k}-5$ 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{($2.7^{k+1}+3. 5^{k+1} -5$)-($2. 7^{k}+3. 5^{k}-5$)}
=2($2. 7^{k}.7+3. 5^{k}.5-5-2. 7^{k}-3. 5^{k}+5$)
=$2(14. 7^{k}+15. 5^{k})-4. 7^{k}-6. 5^{k}$
=$28. 7^{k}+30. 5^{k}- 4. 7^{k}-6. 5^{k}$
=$24. 7^{k}+24. 5^{k}$
=$24( 7^{k}+ 5^{k})$
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.

Answer 2

Hi

answer in attachment...

#tq