Question

${{\text{5}}^{{\text{2n + 2}}}}{\text{ - 24n - 25}}$ is divisible by $576$ for all n ∈ N by using the principle of mathematical induction.
A. True
B. False

Answer 1

According to the principle of mathematical induction law, if the statement is true for n=k where k is a positive integer, then it should be true for n=$k + 1$ also. So in the given question, first we will prove that the statement is true for n=k by first putting n=$1,2,...$ and then k. All the numbers obtained by putting the value of n will be multiple of $576$. Then we will solve the same statement for n=$k + 1$and if it also gives a multiple of $576$then it is true for n=$k + 1$.

Complete step-by-step answer:
We have to find if ${{\text{5}}^{{\text{2n + 2}}}}{\text{ - 24n - 25}}$ is divisible by $576$ for all n ∈ N using the principle of mathematical induction.
Let us assume that P(n) = ${{\text{5}}^{{\text{2n + 2}}}}{\text{ - 24n - 25}}$
Now on putting n=$1$, we get-
$\Rightarrow {\text{P}}\left( 1 \right) = {5^{2 + 2}} - 24 - 25$
On solving, we get-
$\Rightarrow {\text{P}}\left( 1 \right) = {5^4} - 49$
On simplifying, we get-
$\Rightarrow {\text{P}}\left( 1 \right) = 625 - 49 = 576$
So it will be divisible by $576$.
Now on putting n=$2$, we get-
$\Rightarrow {\text{P}}\left( 1 \right) = {5^{2 \times 2 + 2}} - \left( {24 \times 2} \right) - 25$
On solving, we get-
$\Rightarrow {\text{P}}\left( 1 \right) = {5^6} - 48 - 25$
On simplifying further we get,
$\Rightarrow {\text{P}}\left( 1 \right) = 15625 - 73 = 15552 = 576 \times 27$
Hence it is also divisible by $576$.
Let P(n) is true for n=k then we can write-
$\Rightarrow {\text{P(k) = }}{{\text{5}}^{{\text{2k + 2}}}}{\text{ - 24k - 25}}$
On simplifying we get,
$\Rightarrow {\text{P(k) = }}{{\text{5}}^{{\text{2k}}}}^{ + 2}{\text{ - 24k - 25 = 576}}\lambda$ --- (i)
Now let n=$k + 1$then we get-
$\Rightarrow {\text{P(k + 1) = }}{{\text{5}}^{{\text{2}}\left( {{\text{k + 1}}} \right){\text{ + 2}}}}{\text{ - 24}}\left( {{\text{k + 1}}} \right){\text{ - 25}}$
On simplifying we get,
$\Rightarrow {\text{P(k + 1) = }}{{\text{5}}^{{\text{2k + 2}}}}{\text{.}}{{\text{5}}^2}{\text{ - 24k - 24 - 25}}$
On substituting the value of ${5^{2k + 2}}$ from eq. (i) in the above equation, we get-
$\Rightarrow {\text{P(k + 1) = }}\left( {{\text{576}}\lambda {\text{ + 24k + 25}}} \right){{\text{5}}^2}{\text{ - 24k - 24 - 25}}$
On simplifying the above equation, we get-
$\Rightarrow {\text{P(k + 1) = 25}} \times {\text{576}}\lambda {\text{ + 576k - 576}}$
On taking $576$ common, we get-
$\Rightarrow {\text{P(k + 1) = 576}}\left( {{\text{25}}\lambda {\text{ + k - 1}}} \right)$
So we can also write it as-
$\Rightarrow {\text{P(k + 1) = 576}}\nu$ where $\nu$ indicates the multiple of $576$
Hence the obtained number is divisible by $576$so the statement is also true for n=$k + 1$
So we can say it is true for all n ∈ N.
Hence the correct answer is A.

Note: Here the student may make a mistake if they directly assume that P(n) is true for n=k for the given statement. It is necessary to prove that ${\text{P}}\left( 1 \right)$ is true for the given statement, only then can we prove that ${\text{P}}\left( {\text{n}} \right)$ is also true for n=k. Here is ${\text{P}}\left( 1 \right)$ s not true for the given statement then the statement would be false. So first we have to check whether ${\text{P}}\left( 1 \right)$ is true or not.