Question

If the sum of the 3 numbers in g.p. is 31 and their product is 125 find the three numbers

Answer 1

Let the Three Numbers in G.P. be :

$\bf \dfrac{a}{r} \: , \: a \: and\: ar$

Where r is common ratio of corresponding G.P.

According to the Given Condition :

$\bf \dfrac{a}{r} + a + ar = 31 \: \: \: \: . \: .\: .\: \{i \}$

Also ,

$\sf \dfrac{a}{r}.a.ar = 125 \\ \\ \implies \sf {a}^{3} = 125 \\ \\ \sf \implies a = \sqrt[3]{125} \\ \\ \Large \purple{ \implies  \underline {\boxed{{\bf a = 5} }}}$

Putting Value of a in {i} We get ,

$\sf\dfrac{5}{r} + 5 + 5r = 31 \\ \\ \sf\dfrac{5 + 5r + 5 {r}^{2} }{r} = 31 \\ \\ \sf5 {r}^{2} + 5r + 5 = 31r \\ \\ \large \bf5 {r}^{2} - 26r + 5 = 0 \\ \\ \sf {5r}^{2} - 25r - r + 5 = 0 \\ \\ \sf5r(r - 5) - 1(r - 5) = 0 \\ \\ \sf(r - 5)(5r - 1) \\ \\ \Large\bf \purple{ \implies  \underline {\boxed{{\bf r = 5 \: \: \: or \: \: \: r = \frac{1}{5} } }}}$

If r = 5

Numbers will be :

1 , 5 and 25

If r $= \bf\dfrac{1}{5}$

Numbers will be :

25 , 5 and 1

So ,

The three numbers are

$\huge \mathfrak{1,5 \: and \: 25}$

$\Large \red{\mathfrak{  \text{W}hich \:   \: is  \:  \: the  \:  \: required} }\\ \huge \red{\mathfrak{ \text{ A}nswer.}}$