If for x , y ∈ R ,   x > 0 , y = log 10 x + log 10 x 1 / 3 + log 10 x 1 / 9 + … upto…

If for x,y∈R, x>0, y=log10x+log10x1/3+log10x1/9+…upto ∞ terms and 2+4+6+…+2y3+6+9+…+3y=4log10x, then the ordered pair x, y is equal to
  1. 106, 6
  2. 106, 9
  3. 102, 3
  4. 104, 6

Solution

2+4+6+…+2y3+6+9+…+3y=4log10x

21+2+3+----y31+2+3+----y=4log10x

log10x=6

x=106

Now

y=log10x+log10x13+log10x19---∞

=1+13+19---∞log10x

=11-13log10x=9

So, x, y=106, 9

Asked in: JEE Main 2021 (27 Aug Shift 1)

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