Question

three numbers of ap are a,a+d,a+2d there sum is 24 and the sum of the squares is 224. find the term​

Answer 1

Answer:

let the terms be a-d, a, a+d

sum=24

a-d+a+a+d=3a=24

a=8

(a-d) ^2+a^2+(a+d)^2=224

a^2+2(a^2+d^2)=224

3a^2+2d^2=224

3(64)+2d^2=224

2d^2=224-192

d^2=32/2

d=√16=4 put in numbers(assumed )

check calculation.. procedure will b same..

Answer 2

Step-by-step explanation:

Since the numbers are in A.P , they have a common difference 'd' and term 'a'. Also, we have to assume the three numbers to be (a-d), (a), and (a+d) to find out one of the two unknown variables, i.e, a or d

A.T.Q ,

(a-d) (a)+ (a+d) = 24

=》 a -d + a+ a+d = 24

=》 3a + d-d = 24

=》 3a = 24

=》a= 24/3

=》a= 8 ------ eq 1

Also A.T.Q ,

(a-d)^2 + (a)^2 + (a+d)^2 = 224

=》 a^2 -2ad +d^2 + a^2 + a^2 + 2ad +d^2 = 224 { using the formula (a+b)^2 }

=》3a^2 - 2ad + 2ad + d^2 +d^2 = 224

=》3a2 +2d^2 = 224

=》3x8x8 + 2d^2 = 224 ( putting the value of a from eq 1)

=》192 + 2d^2 = 224

=》 2d^2= 224 -192

=》2d^2 = 32

=》d^2 = 16 ( dividing by 2)

=》\/d = \/16 (putting root)

=》 d=4.--------- eq 2

now, we assumed 1st number to be (a-d)

putting the values of a and d, we get = 8-4 = 4

2nd number = a = 8

3rd number =(a+d) = 8+4 =12.

Therefore, the required numbers are 4 , 8 , 12.