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prove that for any natural n the value of the expression (4n+5)^2–9 is divisible by 4.

(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.

366

step-by-step explanation:

34.5 cm

step-by-step explanation:

generally for a circular segment of angle, θ° and radius r

arc length

= (θ/360) x circumference

= (θ/360) x 2πr

in our case, θ = 60°, π=3.142 and r = 33 cm

substitute into above formula

arc length

= (60/360) x 2(3.142)( 33) = 34.5 cm