State the first principle of mathematical induction.
Answer
Let p(n) be a statement involving the natural number n such that:
p(1) is true
p(m + 1) is true, whenever p(m) is true
Then p(n) is true for all $\text{n}\in\text{N}$
This is called first principle of Mathematical Induction.
State the second principle of mathematical induction.
Answer
Let p(n) be a statement involving the natural number n such that:
p(1) is true
p(m + 1) is true, whenever p(m) is true for all $\text{n}\leq\text{m}$
Then p(n) is true for all $\text{n}\in\text{N}$
This is called first principle of Mathematical Induction.
If $p(n): 2 \times 4^{2n-1} + 3^{3n+1} $ is divisible by $\lambda$ for all $\text{n}\in\text{N}$ is true, then find the value of $\lambda$
Answer
$2 \times 42^{n+1}+ 33^{n+1}$
$= 2 \times 24^{n+2} + 3^{3n+1}$
$= 2^{4n+3} + 3^{3n+1}$
Given above expression is divisible by $\lambda$ for all $\text{n}\in\text{N}$
So lets check for $n = 1$
For $n = 2$
$= 2048 + 2187 = 4235$
Now its clearly evident that common factor for above numbers is $11$ so $\lambda = 11$