The value of x for which the numbers 2x+1, 5x+1, and 11x+1 are in G.P is ___ (x is a natural number).

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Solution

If a,b,c are in G.P, then $b^{2}$ = ac

Hence for 2x+1, 5x+1, and 11x+1 to be a G.P

$(5 x + 1)^{2}$ = (2x + 1)(11x + 1)

$25 x^{2}$ + 10x + 1 = $22 x^{2}$ + 13x + 1

$3 x^{2}$ - 3x = 0

3x(x - 1) = 0

x = 0 or x = 1.

Hence, 2x+1, 5x+1, and 11x+1 are in G.P when x = 1.

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