# Solve for x if the numbers are in arithmetic progression . 1+sqrtx, 2-sqrtx,1+2sqrtx.

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1+sqrtx, 2-sqrtx , 1+2sqrtx

Since these are terms of an A.P , then:

(2-sqrtx) - (1+sqrtx) = (1+2sqrtx) - (2-sqrtx)

Now let us simplify:

2 - sqrtx -1 - sqrtx = 1 + 2sqrtx - 2 + sqrtx

Combine like terms:

-sqrtx - sqrtx - 2sqrtx - sqrtx = 1 -2 -2 + 1

==> - 5sqrtx = -2

==> 5sqrtx = 2

==> sqrtx = 2/5

**==> x= 4/25**

The three numbers are in arithmetic progression. We also know that for 3 consecutive terms in an AP the sum of the 1st and 3rd numbers is twice the 2nd number.

Using this 1+ 2 sqrt x + 1 + sqrt x = 2*(2- sqrt x)

=> 2+ 3 sqrt x = 4 - 2 sqrt x

=> 2 = 5 sqrt x

=> sqrt x = 2/5

So x = 4/25

**X is 4/25**

To solve for x in

1+sqrtx, 2-sqrtx,1+2sqrtx which are in AP.

Therefore the common diffrence = 2-sqrtx -(1+sqrtx) = 1+2srtx- (2-srtx)

1 - 2sqrtx = -1 +2sqrtx

Add 2sqrtx :

1 = -1 +4sqrtx.

Add 1:

2 = 4sqrtx

Divide by 2:

1 = 2sqrtx

Square both sides:

1 = 4x

x = 1/4.

If the given numbers are the consecutive terms of an arithmetic series, then the middle terms is the arithmetical average of the previous and subsequent terms.

2-sqrtx = (1+sqrtx+1+2sqrtx)/2

We'll combine like terms:

2-sqrtx = (2+3sqrtx)/2

We'll remove the brackets:

4 - 2sqrtx = 2 + 3sqrtx

We'll add 2sqrtx both sides:

4 = 2 + 5sqrtx

We'll subtract 2 both sides:

4 - 2 = 5sqrtx

2 = 5sqrtx

We'll divide by 5:

sqrtx = 2/5

We'll square raise both sides:

**x = 4/25**