FIND FOUR NUMBERS IN AP WHOSE SUM IS 20 &SUM OF WHOSE SQUARES IS 120
Answers
let four numbers in a.p. be:
a-3d,a-d,a+d,a+3d
their sum=(a-3d)+(a-d)+
(a+d)+(a+3d)=20
=> 4a=20 =>a=5
also sum of squares=120
i.e. (a-3d)2+(a-d)2
+(a+d)2+(a+3d)2=120
{where 2 means square }
=>a2-6ad+9d2-2ad +d2+
a2+2ad+d2+a2+6ad+9d2=120
=>4a2=20d2=120
=>4(25)=20d2=120
=>20d2+20 =>d2=1
=>d=+_1
hence the numbers are
5-3,5-1,5-1,5-3
or 5-3,5-1,5-1,5-3
i.e. 2,4,6,8 or8,6,4,2
Answer:
Let the numbers be a - 3 d , a - d , a + d , a + 3 d .
Sum of the numbers is 20 .
a - 3 d + a - d + a + d + a + 3 d = 20
⇒ 4 a = 20
⇒ a = 20/4
⇒ a = 5
Sum of the squares of the number is 120 .
( a - d )² + ( a - 3d )² + ( a + d)² + ( a + 3d )² = 120
⇒ ( 5 - d )² + ( 5 - 3d )² + ( 5 + d )² + ( 5 + 3d )² = 120
⇒ 25 + d² - 10d + 25 + 9d² - 30d + 25 + d² + 10 d + 25 + 9d² - 30d = 120
⇒ 100 + 20d² = 120
⇒ 20d² = 120 - 100
⇒ 20d² = 20
⇒ d² = 20/20
⇒ d² = 1
⇒ d = ± 1
The A.P will be either :
5 - 3 , 5 - 1 , 5 + 1 , 5 + 3
= 2 , 4 , 6 , 8
or ,
5 + 3 , 5 + 1 , 5 - 1 , 5 - 3
= 8 , 6 , 4 , 2
These are the 2 numbers .
Explanation:
A.P is a sequence where the next term is obtained by adding the previous difference or the common difference . The common difference exists in all the terms .
An A.P will have a constant rate of increase or decrease . The n th term is given by a + ( n - 1 ) d where a is first term , d is the common difference .
The Sum of n terms is n/2 [ 2a + ( n - 1 ) d ]