# if 52 x 54 x 56x --------------- 52n = ( 0.04) - 28 what is n ?

Given

52 $×$ 54 $×$ 56 $×$ ............................... $×$52n    = ( 0.04 )-28

So,

5( 2 + 4 + 6 + ........... + 2n )   = ${\left(\frac{1}{25}\right)}^{-28}$

5( 2 + 4 + 6 + ........... + 2n )  = ( 25 ) 28

5( 2 + 4 + 6 + ........... + 2n )  = 556

Here we can see that ( 2 + 4 + 6 + ........... + 2n ) is a arithmetic progression ,
We know , formula of sum of n terms of arithmetic progression is

S = $\frac{n}{2}$[ 2a+ ( n - 1 ) d ] , Where

a1 = 2 And d = 2  And n = 2n , So

S = $\frac{2n}{2}$[ 4 + ( 2n - 1 ) 2 ]

S = n [ 4 + 4n - 2 ]

S = 4n2 + 2n
So

= 5 56

Comparing both side we get

4n2 + 2n = 56

2n2 +  n - 28 = 0

Now we use quadratic equation to find zeros of the equation is
n
Here a = 2 , b = 1 and c  = -28 So,

So

n = - 4 And  $\frac{7}{2}$                           ( Ans )

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Go to textbook solutions mathematics part 1 ,arithematic sequence,page 25 and question number 3

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