The three numbers a, b, c between $$2$$ and $$18$$ are such that their sum is $$25$$; the numbers $$2$$, a, b are consecutive terms of an A.P. and the numbers b, c, $$18$$ are consecutive terms of a G.P. The three numbers are

Question

The three numbers a, b, c between $$2$$ and $$18$$ are such  that their sum is $$25$$; the numbers $$2$$, a, b are consecutive terms of an A.P. and the numbers b, c, $$18$$ are  consecutive terms of a G.P. The three numbers are

Infinity Learn · Accepted Answer

we have $$a+b+c=25$$       (1)∵$$2$$,a,b are in A.P., ∴ $$2a=2+b$$   (2)∵b,$$c18$$ are in G.P., ∴ $$c2=18b$$    (3)From$$ Eq. $$(1) and (2),   $$3b+2c=48$$; or $$3b=48$$−$$2c$$∴ From Eq. (3),   $$c2=6(48$$−$$2c)=288$$−$$12cor$$  $$c2+12c$$−$$288=0or$$  $$c2+24c$$−$$12c$$−$$288=0$$ $$($$∵$$a=b+12)or$$  $$18b=$$−$$72$$∴$$b=$$−$$4$$,From Eq. (1),  $$a=$$−$$4+12=8$$ Hence, the four numbers are $$8$$, –$$4$$, $$2$$ and $$8$$.

The three numbers a, b, c between 2 and 18 are such that their sum is 25; the numbers 2, a, b are consecutive terms of an A.P. and the numbers b, c, 18 are consecutive terms of a G.P. The three numbers are

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