Question

I will mark BRAINLIEST..............
__________________________

Find four numbers in AP,whose sum is 20 and sum of their squares is 180.

please give a right answer........​

Answer 1

hey mate here is ur answer

let the number be a - 3d, a-d, a+d and a+3d.

thus,

the sum =4a

=>4a=20

=>a=5

also

(5-3d)²+(5+3d)²+(5+d)²+(5-d)=180

=>2(25+9d²)+2(25+d²)=180

=>100+20d²=180

=>5+d²=9

=>d²=4

=>d=±2

hence numbers are

-1,3,7,11

or

11,7,3,-1

alternative method:

let the no be a,a+d,a+2d,a+3d

thus,
a+a+d+a+2d+a+3d=20

4a+6d=20

2a+3d=10

4a²+9d²+12ad=100

4a²+9d²-100=-12ad (equ..1)

also,

a²+(a+d)²+(a+2d)²+(a+3d)²=180

a²+(a²+2ad+d²)+(a²+4ad+4d²)+(a²+6ad+9d²)=180

4a²+12ad+14d²=180

4a²+14d²-180=-12ad (equ..2)

thus,from equ 1 and equ 2

4a²+9d²-100=4a²+14d²-180

5d²=80

d²=16

d=±4

and thus

2a+12=10

2a=-2

a=-1

and hence No's

are -1,3,7,11

Answer 2

Answer:

4.1,4.7,5.3,5.9

Step-by-step explanation:

select the terms as a-3d , a-d , a+d , a+3d their sum is 20

so on solving 4a = 20  i.e a = 5

then sum of their squares is 180

so open all the terms using (a+b)^2

a^2 + 9d^2 - 6ad + a^2 + d^2 - 2ad +a^2 + d^2 + 2ad + a^2 + 9d^2 + 6ad =180

ad terms will cancel out each other

add a^2 separately and d^2 separately 4a^2 + 20d^2 = 180

putting obtained value of a 4*5^2 +20d^2 =180

20d^2 = 180 - 100

d^2 = 80/20

d=+2 or -2

in selected terms put a and d

5-6 , 5-2 , 5+2 , 5+6

they are -1,3,7,11 or 11,7,3,-1

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