Name: Mathematical Induction: Illustration
Uploaded: 2019-07-19
Duration: 4 min 19 s
Description: The formula for the sum of first “n” natural numbers can be derived using the formula of the sum to n terms of AP. Could you think of any other method to der...

Question

Using principle of mathematical induction, Prove that

4^{n + 1} + 15 n + 14

is divisible by 9 for every natural number

n > 0 .

Accepted Answer

To prove - $4^{n + 1} + 15 n + 14$ is divisible by $9$

Step 1 (Verification step) :-

For the least natural number i.e $n = 1$

$4^{1 + 1} + 15 (1) + 14$ is divisible by $9$

$\Rightarrow 16 + 15 + 14$ is divisible by $9$

$\Rightarrow 45$ is divisible by $9$ , which is true.

Thus, the result is true for $n = 1$ .

Step 2 (Assumption step) :-

For $n = k$

Let the result be true for $n = k$ i.e,

$4^{k + 1} + 15 k + 14$ is divisible by $9$

$\Rightarrow 4^{k + 1} + 15 k + 14 = 9 l$

$\Rightarrow 4 (4^{k}) + 15 k + 14 = 9 l \dots \dots (1)$

Step 3 (Induction step) :-

For $n = k + 1$

To prove - $4^{k + 2} + 15 k + 29$ is divisible by $9$

Now

$4^{k + 2} + 15 k + 29$ $= 16 (4^{k}) + 15 k + 29$

$= 16 (\frac{9 l - 15 k - 14}{4}) + 15 k + 29$ {Using $(1)$ }

$= 4 (9 l - 15 k - 14) + 15 k + 29$

$= 36 l - 60 k - 56 + 15 k + 29$

$= 36 l - 45 k - 27$

$= 9 [9 l - 5 k - 3]$ , which is divisible by $9$

Thus, the result is true for $n = k + 1$ .

Step 4 (Generalisation step) :-

The result is true for $n = 1$

And the result is true for $n = k$

This implies the result is true for $n = k + 1$

Therefore, the result is true for every natural numbers $n > 0$ .

Using principle of mathematical induction, Prove that $4^{n + 1} + 15 n + 14$ is divisible by 9 for every natural number $n > 0 .$

Important Questions on Principle of Mathematical Induction

Learn and Prepare for any exam you want !

Using principle of mathematical induction, Prove that 4n+1+15n+14 is divisible by 9 for every natural number n>0.

Important Questions on Principle of Mathematical Induction

Learn and Prepare for any exam you want !

Using principle of mathematical induction, Prove that $4^{n + 1} + 15 n + 14$ is divisible by 9 for every natural number $n > 0 .$