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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 57
Total: 106
If the first and the nth term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.
Solution:  
Let r be the common ratio of the given G.P., then b = nth term = arn−1
⇒ rn−1 =
b
a
 ⇒ r = (
b
a
)
1
n−1
Now, P = product of the first n terms
⇒ P = a.ar.ar2 ... arn−1 = anr1+2+3+...+(n−1) = anr
n(n−1)
2
= an[(
b
a
)
1
n−1
]
n(n−1)
2
 = an(
b
a
)
n
2
 = a
n
2
b
n
2
 = (ab)
n
2
∴ P2 = [(ab)
n
2
]2 = (ab)n
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