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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 29
Total: 106
If
an+bn
an−1+bn−1
 is the A.M. between a and b, then find the value of n.
Solution:  
We have
an+bn
an−1+bn−1
 =
a+b
2
⇒ 2an+2b2 = an+abn−1+an−1b+bn ⇒ an+bn = abn−1+an−1b
⇒ an−abn−1+bn−an−1b = 0 ⇒ a(an−1−bn−1) + b(bn−1−an−1) = 0
⇒ (a - b) (an−1−bn−1) = 0 ⇒ an−1−bn−1 = 0 (Since a - b ≠ 0)
⇒ an−1 = bn−1 ⇒
an−1
bn−1
 = 1 ⇒ (
a
b
)n−1 = 1 = (
a
b
)0
⇒ n - 1 = 0 ⇒ n = 1
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