Math, asked by swroopmalipatil, 9 months ago

The sum of first three terms in an arithmetic progression is 24 and the
sum of their squares is 224. Find the first three terms of this arithmetic
progression.​

Answers

Answered by AnkitBhardwaj420
1

Answer:

Let the roots of A.P .be (a-d) ,a,(a+d)

Now the sum of 1st three terms is 24 that means

a-d+a+a+d=24

3a=24

a=8

we got the value of a=8

Now,

squaring the terms

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

where 2 are square not in multiplication

a2+d2-2ad+a2+a2+d2+2ad=224

3a 2 +2d 2 =224

3×8×8+2d2=224

2d2=224–192

2d2=32

d2=16

d=4

Therefore a-d,a,a+d are

8–4,8,8+4

4,8,12 ←This are the values of 1st three 3terms

Answered by Salmonpanna2022
1

Step-by-step explanation:

Let the first term of the A.P be : a

Let the common difference of the A.P be : d

\longrightarrow  Second term of the A.P will be : a + d

\longrightarrow  Third term of the A.P will be : a + 2d

Given : Sum of first three terms of the A.P is 24

\longrightarrow  a + (a + d) + (a + 2d) = 24

\longrightarrow  3a + 3d = 24

\longrightarrow  3(a + d) = 24

\longrightarrow  a + d = 8

\longrightarrow  a = 8 - d

Given : Sum of squares of the first three terms of the A.P is 224

\longrightarrow  a² + (a + d)² + (a + 2d)² = 224

\longrightarrow  a² + a² + d² + 2ad + a² + 4d² + 4ad = 224

\longrightarrow  3a² + 5d² + 6ad = 224

Substituting the value of a = (8 - d) in the above equation, We get :

\longrightarrow  3(8 - d)² + 5d² + 6d(8 - d) = 224

\longrightarrow  3(64 + d² - 16d) + 5d² + 48d - 6d² = 224

\longrightarrow  192 + 3d² - 48d + 5d² + 48d - 6d² = 224

\longrightarrow  2d² = 224 - 192

\longrightarrow  2d² = 32

\longrightarrow  d² = 16

\longrightarrow  d = ± 4

Consider : d = 4

\longrightarrow  a = (8 - d) = (8 - 4) = 4

\longrightarrow  second term : (a + d) = (4 + 4) = 8

\longrightarrow  Third term : (a + 2d) = (4 + 8) = 12

In this case : The First three terms of the A.P are 4 , 8 , 12

Consider : d = -4

\longrightarrow  a = (8 - d) = (8 + 4) = 12

\longrightarrow  second term : (a + d) = (12 - 4) = 8

\longrightarrow  Third term : (a + 2d) = (12 - 8) = 4

In this case : The First three terms of the A.P are 12 , 8 , 4

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