Question

7^2n+2^3n-3.3^n-1 is divisible by 25

Answer 1

Answer:

The given term $7 ^ { 2 n } + 2 ^ { 3 n - 3 } \cdot 3 ^ { n - 1 }$  is “divisible by 25”.

To find:

Whether the term $7 ^ { 2 n } + 2 ^ { 3 n - 3 } \cdot 3 ^ { n - 1 }$ is divisible by the number 25 or not.

Solution:

Given that

$\begin{array} { c } { 7 ^ { 2 n } + 2 ^ { 3 n - 3 } \times 3 ^ { n - 1 } } \\\\ { \left( 7 ^ { 2 } \right) ^ { n } + \left( 2 ^ { 3 } \right) ^ { ( n - 1 ) } \times 3 ^ { n - 1 } } \\\\ { 49 ^ { n } + 8 ^ { n - 1 } \times 3 ^ { n - 1 } } \\\\ { 49 ^ { n } + 24 ^ { n - 1 } } \\\\ { ( 50 - 1 ) ^ { n } + ( 25 - 1 ) ^ { n - 1 } } \end{array}$

The expanded form is

$\begin{aligned} n c _ { 0 } ( 50 ) ^ { n } - n c _ { 1 } ( 50 ) ^ { n - 1 } \ldots \ldots \ldots & n c _ { n } ( - 1 ) ^ { n } + n c _ { 0 } ( 25 ) ^ { n - 1 } \ldots \ldots - n c _ { n - 1 } ( - 1 ) ^ { n - 1 } \\\\ & 25 k + ( - 1 ) ^ { n } + ( - 1 ) ^ { n - 1 } \end{aligned}$

If n = odd and n-1 = Even

n = Even and n-1 = odd

=25k  

[25 multiplied by some other term, hence it must be divisible by 25 also]

Hence, the given term $7 ^ { 2 n } + 2 ^ { 3 n - 3 } \cdot 3 ^ { n - 1 }$  is “divisible by 25”.