Class - 10 - Mathematics - ARITHMETIC PROGRESSION - October Syllabus - Part - 4

 

Arithmetic Mean

Suppose three quantities a, A, b are in arithmetic progression, then the middle quantity A is said to be arithmetic mean of the two quantities a and b. Since a, A, b are in arithmetic progression.

A = (a + b)/2

Therefore, you can say that the arithmetic mean of two quantities is the half of the sum of those two quantities. Let us understand with the help of some examples:-

Forming an arithmetic progression between two quantities a and b

Exercise - 2

Q.1 Find the arithmetic mean of (1/2) and (-1/2)

Ans.

arithmetic mean  = (1/2) + (-1/2)

= 

= 0

Q.2 Find the arithmetic mean of x² + 3xy and y² – 3xy.

Ans.  arithmetic mean 

= 

 = 

Q.3 Arithmetic mean and product of two numbers are 7 and 45 respectively. Find the numbers.

Ans. Let us consider the two number be x and y.

Given: Arithmetic mean  = 7

(x + y)/2 = 7

x  + y = 14 ....(i)

Product of two number are 45

xy = 45

x = 45/y  ... (ii)

Putting x = 45/y in equation (i)

(45/y) + y = 14

45 + y² = 14y

 y²   -14y +45 = 0

 y²   - 9y - 5y  + 45 = 0

 y(y-9)-5(y-9) = 0

y =9,5

Putting the value of y in equation (ii)

x = 45/9 = 5

x = 45/5 = 9

Hence, the two number are 9 and 5.

Q.4 Arithmetic mean and sum of squares of two numbers are 6 and 90. Find the numbers.

Ans. Let us consider the two number be x and y.

Given: Arithmetic mean  = 6

(x + y)/2 = 6

x + y = 12 ...(i)

sum of squares of two numbers are 90.

x² + y²  = 90 ....(ii)

We now that (x + y)²  =  x² + y²  + 2xy

 (12)²  =  90 + 2xy

144 - 90 = 2xy

64 = 2xy

64/2 = xy 

xy = 32

x = 32/y ...(iii)

Putting x = 32/y in equation(i)

(32/y) + y = 12

32 +  y² = 12y

 y² - 12y + 32 = 0

 y² - 8y - 4y + 32 = 0

y(y -8) - 4(y - 8) = 0

y = 4,8

Putting the value of y in equation(iii)

when y = 4, x = 8

when x = 8, y = 4

Hence, the two number are 9 and 5.

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