Question

Using Binomial Theorem prove that 3 ^ 3n - 26n -1 is divisible by 676.

Answer 1

Answer:

The proof is explained below :

Step-by-step explanation:

$=3^{3\cdot n}-26\cdot n-1\\=27^n-26\cdot n-1\\=(26+1)^n-26\cdot n-1\\\\=\thinspace _{0}^{n}\textrm{C}\cdot 26^n\cdot 1^0 +\thinspace _{1}^{n}\textrm{C}\cdot 26^{n-1}\cdot 1^1 +\thinspace _{2}^{n}\textrm{C}\cdot 26^{n-2}\cdot 1^2 +............+\thinspace _{n-2}^{n}\textrm{C}\cdot 26^2\cdot 1^{n-2}+\\\\\thinspace _{n-1}^{n}\textrm{C}\cdot 26^1\cdot 1^{n-1} +\thinspace _{n}^{n}\textrm{C}\cdot 26^0\cdot 1^n -26\cdot n -1\\\\\text{Now upto }26^{2}\text{ , 676 can be taken out common from each term and the remaining terms}\\\\\text{ can be considered as a constant m}\\\\=676\cdot m+\frac{n!\cdot 26}{(n-1)!}+1-26\cdot n-1\\\\=676\cdot m +n\cdot 26-26\cdot n\\\\=676\cdot m$

which clearly shows the given expression is divisible by 676

Hence Proved.

Answer 2

Answer:

this is the process ....