finding x and don't understand?

Question

wat is x?
1/x + 2/x + 3/x +...+ (x-2)/x + (x-1)/x = 1986
explain plz

Answer 1

Since everything is over the same denominator, you can rewrite it as:

(1 + 2 + 3 + ... + (x - 2) + (x - 1))/x = 1986

Or equivalently,

1 + 2 + 3 + ... + (x - 2) + (x - 1) = 1986x

On the left is the sum of all numbers from 1 to x - 1. There's a formula for this sum: (x - 1)x/2

(x-1)x/2 = 1986x
x(x-1) = 3972x
x² - x = 3972x
x² - 3973x = 0

So either x = 0 (which we have to throw out because it would have 0s in the denominator of the original problem), or

x = 3973

Answer 2

Add up all the fractions you get 1+2+3+ +(x-1), divided by x. But the sum on the top is also x-1 + x-2 + +1. So the double is
x + x + +x that is (x-1) times x. So your initial sum is x(x-1)/2

You divide by x and you want (x-1)/2= 1986 so x-1 = 3972 and x=3973..

Answer 3

Since they all have the same denominator, you can combine the entire numerator into:
1+2+...+x-1

You can use the sum of an arithmetic series:
(x-1)(1+x-1)/2=(x-1)x/2
that is what the numerator equals to.

So (x-1)x/2x=1986
(x-1)/2=1986
x-1=3972
x=3973

Answer 4

Multiplying through by x gives us
1 + 2 + 3 + ... + (x-1) = 1986x

The left-hand side is the sum of the first (x-1) integers (starting at 1).

Recall that 1 + .... + n = n(n+1)/2 - so plug in (x-1) and away you go!

Answer 5

1/x + 2/x + 3/x +...+ (x-2)/x + (x-1)/x = 1986 =
. . . (x - 1)
(1/x)∑ k =
. . . k=1
(1/x)(x - 1)(x)/2 =
(x - 1)/2 = 1986
x = 3,973