Find four numbers in A.P,whose sum is 20,sum of squares is 120?

Question

how can I take the four no,a+d,a_d,a+2d,a-2d or a+d,a-d,a+3d,a-3d why?

Answer 1

arithmatic progression is

a, a+d, a+2d, a+3d

or for convenience we can take

a-3d, a-d, a+d, a+3d (considering increase of "2d")

TIPS: if you have to find count of odd numbers, u take the increase of "d" and for even numbers, u take the increase of "2d"
it gives us, an easier calculation, as shown in the addition and squarring steps of this problem.

now addition of 4 numbers is 20

[a-3d] + [a-d] + [a+d] + [a+3d] = 20

4a = 20 (-3d, -d, +d, +3d cancels each other)

a = 20/4

a= 5 (A)

now, sum of squares is 120

(a-3d)**2 + (a-d)**2 + (a+d)**2 + (a+3d)**2 = 120 (I used ** for squaring or exponentiating)

[a**2 + 9d**2 - 6ad] + [a**2 + d**2 - 2ad] + [a**2 + d**2 + 2ad] + [a**2 + 9d**2 + 6ad] = 120

4a**2 + 20d**2 = 120 (-6ad -2ad +2ad + 6ad cancels each other)
4[5]**2 + 20d**2 = 120 use (A)
4[25] + 20d**2 = 120
100 + 20d**2 = 120
20d**2 = 120-100
20d**2 = 20
d**2 = 20/20
d**2 = 1
d=+1 or -1
in AP the difference cannot be negative.
hence d = 1

now the numbers are

a-3d, a-d, a+d, a+3d

5-3, 5-1, 5+1, 5+3

2,4,6,8

GOOD LUCK

Answer 2

2 4 6 8

Answer 3

2 4 6 8

Answer 4

Well, 11^2, by itself, is 121, so all the numbers have to be 10 or less, so that helps.... 8 + 2 + 4 + 6 = 20 64 + 4 + 16 + 36 = 120 2,4,6,8

Answer 5

Ans 2 4 6 8 (U JUST SOLVE IT BY TAKING THE NUMBERS AS a-2d,a-d,a,a+d) it"s quite a simple job
u can solve it by using a-3d,a-d,a+d,a+3d but the job gets a bit clumsier

Answer 6

2, 4, 6, 8

Answer 7

Answer is 2,4,6,8

Let the least number be a and the c.d. be k
So the 4 numbers of AP are a, a+k, a+2k, a+3k
Given: a + (a+k) + (a+2k) + (a+3k) = 20 ---(i)
and a^2 + (a+k)^2 + (a+2k)^2 + (a+3k)^2 = 120 ---(ii)
Sol: From (i), 2a + 3k = 10 --(iii)
From (ii), 4a^2 + 14k^2 + 6ka = 120 --(iv)
Squaring both sides of (iii) => 4a^2 + 9k^2 + 12ka = 100 -- (v)

(iv) - (v) => k^2 = 4
So, k =2 or -2

This gives series as 2,4,6,8
or 8,6,4,2

Answer 8

2,4,6,8

2+4+6+8=20
2^2=4
4^2=16
6^2=36
8^2=64
4+16+36+64=120

Answer 9

u take d no.'s in AP as a-3d,a-d,a+d,a+3d
add all these no.'s & equate By 20.1st equatin
add d squares of individuals & equate By 120.2nd equation
after solving these 2 equations u get d result.


result is 2,4,6,8

Answer 10

you can take
a,a+d,a+2d,a+3d

or
a,a-d,a-2d,a-3d

let me choose the first case
then 2a+3d=10
and
2a^2+7d^2+6ad=60
find a, d