What is ((k!)^2)/(((k+1)!)^2) ?

The solution to ((k!)^2)/(((k+1)!)^2) is 1/((k+1)^2)

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simplify ((k!)^2)/(((k+1)!)^2)

simplify

\frac{( k ! ) ^{2}}{( ( k + 1 ) ! ) ^{2}}

\frac{1}{( k + 1 ) ^{2}}

Solution steps

\frac{k ! ^{2}}{( k + 1 ) ! ^{2}}

Apply exponent rule:

a^{2} = aa

k!^{2} = k! k!

= \frac{k ! k !}{( k + 1 ) ! ^{2}}

Apply exponent rule:

a^{2} = aa

(k + 1)!^{2} = (k + 1)! (k + 1)!

= \frac{k ! k !}{( k + 1 ) ! ( k + 1 ) !}

Cancel the factorials:

\frac{n !}{( n + m ) !} = \frac{1}{( n + 1 ) \cdot ( n + 2 ) \dots ( n + m )}

\frac{k !}{( k + 1 ) !} = \frac{1}{( k + 1 )}

= \frac{1}{( k + 1 ) ( k + 1 )}

Apply exponent rule:

aa = a^{2}

(k + 1) (k + 1) = (k + 1)^{2}

= \frac{1}{( k + 1 ) ^{2}}

x^{2} = 22 + 10 x

2 \cdot 2 x^{\frac{5}{4}} - 2 x^{\frac{1}{4}}

x^{2} - 6 x + 8 \geq 0

\frac{5 ( - 4 ( 4 ) ^{4} + 75 ( 4 ) ^{2} )}{25 - ( 4 ) ^{2}}

What is ((k!)^2)/(((k+1)!)^2) ?
The solution to ((k!)^2)/(((k+1)!)^2) is 1/((k+1)^2)