1
You visited us
1
times! Enjoying our articles?
Unlock Full Access!
Byju's Answer
Standard XI
Mathematics
Absolute Value Function
If a, b, c ar...
Question
If a, b, c are in A.P., prove that:
(i) (a − c)
2
= 4 (a − b) (b − c)
(ii) a
2
+ c
2
+ 4ac = 2 (ab + bc + ca)
(iii) a
3
+ c
3
+ 6abc = 8b
3
.
Open in App
Solution
Since a, b, c are in A.P., we have:
2b = a+c
⇒
b =
a
+
c
2
(i) Consider RHS:
4 (a − b) (b − c)
Substituting
b
=
a
+
c
2
:
⇒
4
a
-
a
+
c
2
a
+
c
2
-
c
⇒
4
2
a
-
a
-
c
2
a
+
c
-
2
c
2
⇒
a
-
c
a
-
c
⇒
a
-
c
2
Hence, proved.
(ii) Consider RHS:
2 (ab + bc + ca)
Substituting
b
=
a
+
c
2
:
⇒
2
a
a
+
c
2
+
c
a
+
c
2
+
a
c
⇒
2
a
2
+
a
c
+
a
c
+
c
2
+
2
a
c
2
⇒
a
2
+
4
a
c
+
c
2
Hence
,
proved
.
(iii) Consider RHS:
8
b
3
Substituting
b
=
a
+
c
2
:
⇒
8
a
+
c
2
3
⇒
a
3
+
c
3
+
3
a
c
a
+
c
⇒
a
3
+
c
3
+
3
a
c
(
2
b
)
⇒
a
3
+
c
3
+
6
a
b
c
Hence, proved.
Suggest Corrections
0
Similar questions
Q.
If a,b,c are in AP, prove that :
(i)
(
a
−
c
)
2
=
4
(
a
−
b
)
(
b
−
c
)
(ii)
a
2
+
c
2
+
4
a
c
=
2
(
a
b
+
b
c
+
c
a
)
Q.
If
a
,
b
,
c
are A.P prove that:
a
2
+
c
2
+
4
a
c
=
2
(
a
b
+
b
c
+
c
a
)
Q.
If
a
,
b
,
c
are in A.P prove that:
(i)
(
a
−
c
)
2
=
4
(
a
−
b
)
(
b
−
c
)
Q.
If
a
,
b
,
c
are in A.P prove that:
a
3
+
c
3
+
6
a
b
c
=
8
b
3
Q.
Factorise
a
3
+
8
b
3
+
c
3
−
6
a
b
c
using the identity
a
3
+
b
3
+
c
3
−
3
a
b
c
=
1
2
(
a
+
b
+
c
)
[
(
a
−
b
)
2
+
(
b
−
c
)
2
+
(
c
−
a
)
2
]
View More
Join BYJU'S Learning Program
Grade/Exam
1st Grade
2nd Grade
3rd Grade
4th Grade
5th Grade
6th grade
7th grade
8th Grade
9th Grade
10th Grade
11th Grade
12th Grade
Submit
Related Videos
Modulus
MATHEMATICS
Watch in App
Explore more
Absolute Value Function
Standard XI Mathematics
Join BYJU'S Learning Program
Grade/Exam
1st Grade
2nd Grade
3rd Grade
4th Grade
5th Grade
6th grade
7th grade
8th Grade
9th Grade
10th Grade
11th Grade
12th Grade
Submit
Solve
Textbooks
Question Papers
Install app