The sum of the 3 number is 14 and the sum of squares of the number is 100,the product of the number is 24 find the sum of the number
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The sum of cubes of the numbers = 19700
Step-by-step explanation:
The sum of three numbers is 14
Sum of their square is 100
Their product is 24
TO DETERMINE
The sum of cubes of the numbers
EVALUATION
Let the numbers are a , b , c
By the given condition
a + b + c = 14
a² + b² + c² = 1000
abc = 24
Now
a² + b² + c² - ab - bc - ca
= 1/2 ( 2a² + 2b² + 2c² - 2ab - 2bc - 2ca )
= 1/2 [ 3( a² + b² + c² ) - ( a² + b² + c² + 2ab + 2bc + 2ca ) ]
= 1/2 [ 3( a² + b² + c² ) - ( a + b + c ) ² ]
= 1/2 [ 3000 - 14² ]
= 1/2 [ 3000 - 196 ]
= 1/2 × 2804
= 1402
Now
a³ + b³ + c³ - 3abc = ( a + b + c )(a² + b² + c² - ab- bc - ca )
⇒ a³ + b³ + c³ - 3 × 24 = 14 × 1402
⇒ a³ + b³ + c³ - 72 = 19628
⇒ a³ + b³ + c³ = 19700
FINAL ANSWER
The sum of cubes of the numbers = 19700
Answer from: Pulakmath007
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