How does:
(k(k-1))/2! * (1/k)^2
simplify into:
(1- (1/k))/2!
Could you show me step-by-step? thanks.
(k(k-1))/2!*(1/k)^2 square 1/k and group into one single faction
= (k(k-1))/(2!*(k)^2) the k on top and the k on bottom cancel
= (k-1)/(2!*k) you can rewrite this as:
= (1/2!)*((k-1)/k) now multiply the whole thing by (1/k)/(1/k) {note: this is legal because it is the same as multiplying a number by 1}
= (1/2!)*((1/k)(k-1))/((1/k)*k) the bottom cancel out {1/k*k = 1} and just multiply the top out
= (1/2!)*(1-(1/k)) rewrite as:
= (1-(1/k))/2!
Fun problem!
What happened to the k in the 2!*k before you multiplied everything by (1/k)/(1/k)
(k(k-1))/2! * (1/k)^2
= (k²-k)/k² / 2!
= (k²/k² - k/k²) /2!
= (1 - 1/k)/2!
To simplify the expression (k(k-1))/2! * (1/k)^2, let's break it down step-by-step:
Step 1: Simplify the factorial expression
The term 2! means 2 factorial, which is equal to 2 x 1.
So, 2! = 2 x 1 = 2.
Step 2: Expand the expression using distributive property
(k(k-1))/2! * (1/k)^2 = (k(k-1))/2 * (1/k)^2.
Step 3: Simplify the expression (1/k)^2
To simplify (1/k)^2, square the numerator and square the denominator:
(1/k)^2 = (1^2)/(k^2) = 1/k^2.
Step 4: Substitute the simplified expression into the original equation
Substituting 1/k^2 back into the original equation, we have:
(k(k-1))/2 * (1/k)^2 = (k(k-1))/2 * (1/k^2).
Step 5: Simplify the expression (k(k-1))/2
To simplify the expression (k(k-1))/2, multiply the terms in the numerator:
(k(k-1)) = k^2 - k.
So, (k(k-1))/2 = (k^2 - k)/2.
Step 6: Substitute the simplified expression back into the original equation
Substituting (k^2 - k)/2 back into the equation, we have:
(k(k-1))/2 * (1/k^2) = ((k^2 - k)/2) * (1/k^2).
Step 7: Simplify the expression ((k^2 - k)/2) * (1/k^2)
To simplify the expression ((k^2 - k)/2) * (1/k^2), multiply the terms in the numerators and denominators:
((k^2 - k)/2) * (1/k^2) = (k^2 - k)/(2k^2).
Step 8: Factor out k from the numerator
To further simplify the expression (k^2 - k)/(2k^2), factor out k from the numerator:
(k^2 - k) = k(k - 1).
So, (k^2 - k)/(2k^2) = (k(k - 1))/(2k^2).
Step 9: Simplify the expression (k(k - 1))/(2k^2)
To simplify (k(k - 1))/(2k^2), divide k from the numerator and the denominator:
(k(k - 1))/(2k^2) = (k - 1)/(2k).
Therefore, the expression (k(k-1))/2! * (1/k)^2 simplifies to (k - 1)/(2k).