find three number in G.P. such that their sum is 21 and sum of their squre is 189 ?
Answers
Answer:
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Answer:
Let three numbers in G.P be a,ar,ar
2
Since the sum of the numbers is 21
∴a+ar+ar
2
=21
⇒a(1+r+r
2
)=21 .....(1)
Since the sum of the squares of the numbers is 189
∴a
2
+a
2
r
2
+a
2
r
4
=189
⇒a
2
(1+r
2
+r
4
)=189 ......(2)
Squaring both sides of (1), we get
a
2
(1+r+r
2
)
2
=441 .......(3)
Dividing (3) by (2), we get
a
2
(1+r
2
+r
4
)
a
2
(1+r+r
2
)
2
=
189
441
⇒
(1+r
2
+r
4
)
(1+r+r
2
)
2
=
9
21
⇒
(1+2r
2
+r
4
−r
2
)
(1+r+r
2
)
2
=
3
7
⇒
(r
2
+1)
2
−r
2
(1+r+r
2
)
2
=
3
7
⇒
(r
2
+1−r)(r
2
+1+r)
(1+r+r
2
)
2
=
3
7
⇒
(r
2
+1−r)
(1+r+r
2
)
=
3
7
⇒3r
2
+3r+3=7r
2
−7r+7
⇒4r
2
−10r+4=0
⇒2r
2
−5r+2=0
⇒r=
2×2
5±
5
2
−4×2×2
=
4
5±
25−16
=
4
5±
9
=
4
5±3
=
4
8
,
4
2
=2,
2
1
When r=2 from (1)
a(1+2+2
2
)=21⇒a=3
∴ Numbers are 3,3×2,3×2
2
or 3,6,12
When r=
2
1
from (1)
a[1+
2
1
+
4
1
]=21
⇒a[
4
4+2+1
]=21
⇒a=
7
21×4
=3×4=12
∴ Numbers are 12,12×
12
1
,12×(
12
1
)
2
or 12,6,3