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Question

Let xn,yn,zn,wn denote nth term of four different arithmetic progressions with positive terms. If x4+y4+z4+w4=8 and x10+y10+z10+w10=20, then maximum possible value of x20y20z20w20 is

A
104
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B
108
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C
1010
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D
1020
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Solution

The correct option is A 104
Let xn=ax+(n1)dx
yn=ay+(n1)dy
zn=az+(n1)dz
wn=aw+(n1)dw

Given x4+y4+z4+w4=8
ax+ay+az+aw+3(dx+dy+dz+dw)=8(1)

Also x10+y10+z10+w10=20
ax+ay+az+aw+9(dx+dy+dz+dw)=20(2)

Solving equation (1) and (2). we get
dx+dy+dz+dw=2ax+ay+az+aw=2

As the terms of the A.P. is positive, using A.MG.M, we get
x20+y20+z20+w204(x20y20z20w20)1/4ax+ay+az+aw+19(dx+dy+dz+dw)4(x20y20z20w20)1/410(x20y20z20w20)1/4x20y20z20w20104

Hence, the maximum possible value of the given expression is 104.

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