three numbers are in arthimethic progresion their sum is 7.5 and their product is 10.find the numbers?

Answer 1

If some numbers are in arithmetic progression, that means that the difference between any two consecutive numbers is a constant.
Let's assume that:
a1, a2, a3 -> the 3 numbers
r -> the ration (r=a2-a1=a3-a2)
Thusly, we have:
a2 = a1+r
a3 = a2+r = a1+r+r = a1 + 2r
(In general we have: an= a1+(n-1)r)
Since their sum is 7.5 => a1+(a1+r)+(a1+2r) = 7.5 =>
3(a1+r) = 7.5 => a1+r = 2.5 => a2 = 2.5

Now, since their product is 10 => a1*a2*a3 = 10 => a1*2.5*(a1+2r) = 10 => a1*(a1+2r) = 4 => (a2-r)(a2+r) = 4 => a2^2-r^2 = 4 => (2.5)^2-r^2 = 4 => 6.25 - r^2 = 4 => r^2 = 2.25 => r=+-1.5
Now, you'll notice we have both a positive and a negative value for r. Don't worry though, this is normal, since we didn't specify which one of the 3 numbers is the greatest.
If a1>a2>a3 => r=-1.5
Otherwise, if a1 r=1.5
It really doesn't matter though, since the numbers will stay the same, they'll just be in a different order.
Since you now know a2 and r, you can figure out the other 2 numbers:
a1 = a2-r = 2.5-1.5 = 1
a3 = a2+r = 2.5+1.5 = 4

You might want to check you results, just to make sure:
a1+a2+a3 = 1+2.5+4 = 7.5
a1*a2*a3 = 1*2.5*4 = 10, therefor the numbers we just found are correct.
Hope you understood the method. It's really easy once you get the hang of it

Answer 2

Arithmetic progression means they have common difference, Let the difference be 'd' then let the numbers be
(n–d) + n + (n+d) = 7.5
Therefore 3n = 7.5
n = 2.5
This mean the middle number is 2.5

(n– d)n(n+d) = 10
But n = 2.5 .
Therefore 2.5(n² – d²) = 10
n² – d² = 4
6.25 – d² = 4
d² = 2.25
d = + 1.5 or – 1.5
This gives three Numbers in arithmetic progression as 1, 2.5 and 4 (when d = + 1.5)
or 4, 2.5, 1 (when d = – 1.5)

Answer 3

Expression of three numbers in arithmetic progression can be presented as
(n-d) + n + (n+d) = 7.5
Therefore 3n = 7.5
This gives us n = 2.5
This mean the middle number is 2.5
To find out the difference between two numbers in arithmetic progression we have
(n-d)n(n+d) = 10
But n = 2.5 from previously solved equation.
Therefore 2.5(n² - d²) = 10
n² - d² = 4
6.25 - d² = 4
d² = 2.25
d = + or - 1.5
This gives three Numbers in arithmetic progression as 1, 2.5 and 4

Answer 4

a million) 2 + 4 + 6 = 12 2) 5 cases the smallest numbers equals the sum of the different 2. for this reason, the 1st quantity is two 3) the biggest quantity equals the sum of the different 2. for this reason, the biggest quantity is 6. No different quantity than 4 could be utilized because of the fact the final quantity interior the set, for this reason the respond is: 2, 4, 6

Answer 5

x + (x + n) + (x + 2n) = 7.5
x(x + n)(x + 2n) = 10

3x + 3n = 7.5
3n = 7.5 - 3x
n = (7.5 - 3x) / 3

(x^2 + nx)(x + 2n) = 10
x^3 + nx^2 + 2nx^2 + 2(n^2)x = 10
x^3 + 3nx^2 + 2(n^2)x - 10 = 0

x^3 + 3((7.5 - 3x) / 3) x^2 + 2((7.5 - 3x) / 3)^2 * x - 10 = 0
x^3 + 7.5x^2 - 3x^3 + 2x((7.5^2 - 45x + 9x^2)/9) - 10 = 0
x^3 + 7.5x^2 - 3x^3 + 12.5x - 10x^2 + 2x^3 - 10 = 0
12.5x - 2.5x^2 - 10 = 0
25x - 5x^2 - 20 = 0
5x^2 - 25x + 20 = 0
x^2 - 5x + 4 = 0
(x - 1)(x - 4) = 0

x = 1
or
x = 4

n = (7.5 - 3x) / 3

If x = 1
n = (7.5 - 3) / 3
n = 1.5

If x = 4
n = (7.5 - 4) / 3
n = 7/6

If x = 1
The numbers are 1, 2.5, and 4

If x = 4
The numbers are 4, 31/6, 19/3
The sum of the progression is greater than 7.5, so this solution is extraneous.

So the numbers are 1, 2.5, and 4

Answer 6

Let the three numbers be (a-d,a,(a+d.)
their sum =3a=7.5, a=2.5
their product=a(a^2-d^2)=10
a^2-d^2=4, where we used a=2.5.
d^2=6.25-4=2.25
d=1.5, so the three numbers required are 1,2.5,4. ANS.

Answer 7

1, 2.5, 4?
where the arithmetic progression can be represented as
a(n) = a(n-1) + 1.5, where a(1) = 1

That's just a guess though, I don't know how to solve it arithmetically.

Answer 8

(n-d) + n + (n+d) = 7.5
3n = 7.5
n = 2.5

(n-d)n(n+d) = 10
2.5(n² - d²) = 10
n² - d² = 4
6.25 - d² = 4
d² = 2.25
d = 1.5

numbers are 1, 2.5, 4