Question

find two whose arithmetic mean is 18.5 and geometric mean is 17

Answer 1

Answer:

We know that the arithmetic mean between the two numbers a and b is AM=

2

a+b

and the geometric mean is GM=

ab

.

Here, it is given that the arithmetic mean of a and b is 17, therefore,

AM=

2

a+b

⇒17=

2

a+b

⇒17×2=a+b

⇒a+b=34...(1)

Also, it is given that the geometric mean of a and b is 15, therefore,

GM=

ab

⇒15=

ab

⇒ab=15

2

⇒ab=225.....(2)

Now consider (a−b)

2

and use equations 1 and 2 as follows:

(a−b)

2

=(a+b)

2

−4ab=(34)

2

−(4×225)=1156−900=256 implies that

a−b=16.....(3)

Adding equations 1 and 3, we have

(a+a)+(b−b)=34+16

⇒2a=50

⇒a=

2

50

=25

Substitute the value of a in equation 1 as follows:

25+b=34

b=34−25=9