Question

Find five numbers in G.P. such that their product is 243 and the sum of the second and fourth numbers is 10.

Answer 1

Let the five numbers in G.P. be

\(\frac{a}{r^2},\frac{a}r, a, ar, ar^2.\)

According to the given condition,

When a = 3, r = 3,

\(\frac{a}{r^2}=\frac13, \frac{a}{r} =1\), ar = 9, ar² = 27

Hence, the five numbers in G.P. are

27, 9, 3,1, 1/3 or 1/3, 1, 3, 9, 27

Answer 2

Let the 5 numbers be

a/r2, a/r, ,a, ar, ar2,

So their product a5 =243

Hence a =3

Again sum of  2nd and 4th numbers

a/r+ar = 10

=> 3/r+3r=10

=>3r2-10r+3=0

=>3r2 -9r-r+3=0

=(r-3)(3r-1)=0

So r=3 or 1/3

Hence the 5 numbers are

1/3,1,3,9,27

Or 27,9,3,1,1/3