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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 61
Total: 106
Find the value of n so that
an+1+bn+1
an+bn
 may be the geometric mean between a and b.
Solution:  
If
an+1+bn+1
an+bn
 is G.M. between a and b, Then
an+1+bn+1
an+bn
 = √ab
⇒ an+1+bn+1 = (an+bn)√ab ⇒ an+1+bn+1 = an+
1
2
b
1
2
+a
1
2
bn+
1
2
⇒ an+1−an+
1
2
b
1
2
 = a
1
2
bn+
1
2
−bn+1
⇒ an+
1
2
(a
1
2
−b
1
2
) = a
1
2
bn+
1
2
−bn+1
⇒ an+
1
2
(a
1
2
−b
1
2
) = bn+
1
2
(a
1
2
−b
1
2
)
⇒ an+
1
2
 = bn+
1
2
 ⇒ (
a
b
)n+
1
2
 = 1 = (
a
b
)0 ⇒ n +
1
2
 = 0 ⇒ n = −
1
2
Hence the value of n is −
1
2
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