Answer:
(4n+5)² - 9 = 4 ( 4n² + 4 + 2n)
since, we have a factor 4 common from the expression
therefore, we can say that for any natural n the value of the expression (4n+5)^2–9 is divisible by 4.
Step-by-step explanation:
Data provided in the question:
Expression : (4n+5)² - 9
To prove: for any natural n the value of the expression (4n+5)² - 9 is divisible by 4.
Now,
This can be proved if there is on factor of 4 that can be taken from the expression
Thus,
solving the expression
(4n+5)² - 9
= [ (4n)² + 5² + 2 × 4n × 5 ] - 9 [because ( a + b )² = a² + b² + 2ab]
or
= 16n² + 25 + 8n - 9
or
= 16n² + 16 + 8n
now taking 4 common from the above expression, we get
= 4 ( 4n² + 4 + 2n)
since, we have a factor 4 common from the expression
therefore, we can say that for any natural n the value of the expression (4n+5)^2–9 is divisible by 4.
