.find 10 arithmetic means between 5 and "-17" and prove that the sum of all 10 AM is ten times the A. M between them.
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Answer:
Let, a and b two given numbers.
Let,
A
1
,A
2
,A
3
.A
n
be Arithmetic means between a and b.
a,A
1
,A
2
,A
3
.A
n
,b are in A.P.
a+A
1
+A
2
+A
3
+A
n
+b=(
2
n+2
)(a+b) (first term is =a, last term=b, number of term=n+2)
A
1
,A
2
,A
3.A
n
=
2
(n+2)(a+b)
−(a+b)
=(a+b)[
2
n+2
−1]
=(a+b)[
2
n+2−2
]
=n(
2
a+b
)
= n times the single A.M. Between a and b
Hence, this is the answer.
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