Line 130: |
Line 130: |
| a + b – 10b = 0<br> | | a + b – 10b = 0<br> |
| a – 9b = 0<br> | | a – 9b = 0<br> |
− | a = 9b<br>
| + | a = 9b<br> |
| Consider equation 1<br> | | Consider equation 1<br> |
| a + b = 2 (<math>\sqrt{ab}</math> + 2 )<br> | | a + b = 2 (<math>\sqrt{ab}</math> + 2 )<br> |
Line 149: |
Line 149: |
| a + 1 = 10<br> | | a + 1 = 10<br> |
| a = 10-1<br> | | a = 10-1<br> |
− | a = 9 | + | a = 9<br> |
| + | |
| =Problem 4= | | =Problem 4= |
| ''' Exercise 3.7 , Problem number 10, Page number 62 ''' | | ''' Exercise 3.7 , Problem number 10, Page number 62 ''' |
Line 164: |
Line 165: |
| #In this problem we prove that 'c' will be the harmonic mean between 'a' and 'b' | | #In this problem we prove that 'c' will be the harmonic mean between 'a' and 'b' |
| #That is we just show that c = <math>\frac{2ab} {a + b}</math> | | #That is we just show that c = <math>\frac{2ab} {a + b}</math> |
| + | ==Solution of the problem== |
| + | Assumption :- |
| + | #He know the formula for A.M , G.M and H.M |
| + | #He know the basic operation. |
| + | #He know substitute the values |
| + | ==Algorithm== |
| + | The formula for arithmetic mean between 'a' and 'b' is A.M = <math>\frac{a + b} {2}</math><br> |
| + | The formula for geometric mean between 'a' and 'b' is G.M = <math>\sqrt{ab}</math><br> |
| + | Then formula for harmonic mean between 'a' and 'c' is H.M = <math>\frac{2ab} {a + b}</math><br> |
| + | Let 'a' be the arithmetic mean of 'b' and 'c'<br> |
| + | That is a = <math>\frac{b + c} {2}</math><br> |
| + | Re-arrange the formula,<br> |
| + | 2 = <math>\frac{b + c} {a}</math><br> |
| + | Multiply both side by 'ab' we get,<br> |
| + | 2ab = <math>\frac{ab(b + c)} {a}</math><br> |
| + | Cancel 'a' in Right hand side and multiply 'b' inside to the bracket we get<br> |
| + | 2ab = <math>b^{2}</math> + bc ---------->1<br> |
| + | Also 'b' is the geometric mean between 'a' and 'c'<br> |
| + | That is b = <math>\sqrt{ac}</math><br> |
| + | We also write this as <math>b^{2}</math> = ac.-------->2<br> |
| + | Now substitute thia value In equation 1,<br> |
| + | 2ab = ac + bc<br> |
| + | Take common in right hand side ( c is common )<br> |
| + | 2ab = c(a + b)<br> |
| + | Divide both side by (a + b),<br> |
| + | <math>\frac{2ab} {a + b}</math>= c<br> |
| + | Hence 'c' is the harmonic between 'a' and 'b'. |
| + | |
| + | [[Category:Progressions]] |