Question

find three numbers in GP whose sum is 35 and the sum of their squares is 525​

Answer 1

5 , 10 , 20 are the numbers in GP whose sum is 35 and the sum of their squares is 525​

Step-by-step explanation:

Let say three numbers are

a  ar  ar²

a + ar + ar² = 35

=>a (1 + r + r²) = 35

Squaring both sides

=> a² (1 + r + r²)² = 1225

a² + (ar)² + (ar²)²  = 525

=> a² + a²r² + a²r⁴ = 525

=> a²(1 + r² + r⁴)  = 525

a²(1 + r + r²)²/ a²(1  + r² + r⁴)   = 1225/525

=> (1 + r + r²)²/ (1  + r² + r⁴)   = 7/3

=> 3(1 + r + r²)² = 7(1  + r² + r⁴)

=> 3( r⁴ + 2r³ + 3r² + 2r + 1) = 7(1  + r² + r⁴)

=> 4r⁴ - 6r³ -2r² -6r + 4 = 0

=> 2r⁴ - 3r³ - r² - 3r + 2 = 0

=> (r - 2)(2r³ + r² + r - 1) = 0

=> (r - 2)(2r - 1)(r² + r + 1) = 0

=> r = 2 or 1/2

a(1 + r + r²) = 35

=> a(1 + 2 + 4) =35  => a = 5

5 , 10 , 20

or

a(1 + 1/2 + 1/4) = 21 => 7a/4 = 35 => a = 20

20 , 10 , 5

5 , 10 & 20 are three numbers in GP whose sum is 35 and the sum of their squares is 525​

Learn more :

Find three numbers in GP such that their sum is 35 and their products is 1000​

https://brainly.in/question/13158652

https://brainly.in/question/12377053

Answer 2

The G.P is 5,10,20.

The G.P is 20,10,5.

Step-by-step explanation:

Let the three numbers in G.P are $a,ar,ar^2$

The sum of three terms of G.P is 35.

$a + ar + ar^2 = 35$

$a(1+r+r^2)= 35$ ....(1)

The sum of their squares is 525.

$a^2 + (ar)^2 + (ar^2)^2 =525$

$a^2 +a^2r^2+a^2r^4=525$

$a^2(1+r^2+r^4)=525$ ....(2)

Squaring equation (1) both side,

$a^2(1+r+r^2)^2=35^2$

$a^2(1+r+r^2+r^4)+a^2(2r)(1+r+r^2)=1225$

$a^2(1+r+r^2+r^4)+a^2(2r)(1+r+r^2)=1225$

$525+2ar(35)=1225$

$70ar=1225-525$

$ar=\frac{700}{70}$

$ar=10$

$a=\frac{10}{r}$

Substitute the value in (1),

$\frac{10}{r}(1+r+r^2)= 35$

$10+10r+10r^2=35r$

$10r^2-25r+10=0$

$2r^2-5r+2=0$

$2r^2-4r-r+2=0$

$2r(r-2)-1(r-2)=0$

$(r-2)(2r-1)=0$

$r=2,\frac{1}{2}$

When r=2,

$a=\frac{10}{2}=5$

The G.P is 5,10,20.

When $r=\frac{1}{2}$,

$a=\frac{10}{\frac{1}{2}}=20$

The G.P is 20,10,5.

#Learn more

Geometric progression

brainly.in/question/13158652, Answered by ParthYadav10056.