Question

find the three terms in G.P whose product is 729 and sum of their product in pairs is 351

Answer 1

Solution :-

Let the three numbers be a, ar, ar².

a³r³ = 729

ar = 9

a(ar)+ar(ar²)+a(ar²) = 351

a²r + a²r³ + a²r² = 351

81/r + 81r + 81 = 351

81 (1/r + r + 1 ) = 351
Now , divide both side by 81

1/r + r + 1= 351/81
1/r + r + 1 = 13/3
Now , we will subtract 1 both sides
1/r + r + 1 - 1 = 13/3 - 1
1/r + r = 10 /3
(1+ r ^2) / r = 10/ 3

Now , we will do cross multiplication

3( 1+ r^2 ) = 10 r
3+ 3r^2 = 10 r
3r^2 -10 r + 3 = 0
3r ^2 - 9r - r + 3 = 0
3r ( r - 3 ) - 1 ( r - 3) = 0

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

So r = 3 , r =1/3

Now , we will find a

If r = 3 and we had ar = 9

Now we will plug r = 3 in ar = 9

a * 3 = 9

3a = 9

So a= 3

We had three numbers a , ar , ar ^2

After plugging the value of a and r , So we get 3 , 3 * 3 = 9 , 3 * (3) ^2 = > 3 . 9 = > 27

We will get same number if we will plug r = 1/3 in ar = 9 , we will get a = 27 , So we will get same numbers

Hence the three numbers are, 3 , 9 , 27 answer.