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Question

If the arithmetic mean of a&bis an+bnan-1+bn-1, then the value of n is:


A

-1

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B

0

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C

1

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D

None of these

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Solution

The correct option is C

1


Explanation for the correct option.

Step 1: Simplify the expression of mean
It is given that AM between a&b=an+bnan-1+bn-1.

We know that the arithmetic mean between a&b is: AM=a+b2

⇒an+bnan-1+bn-1=a+b2⇒2an+2bn=aan-1+bn-1+ban-1+bn-1⇒2an+2bn=aan-1+abn-1+ban-1+bbn-1⇒2an+2bn=an+abn-1+ban-1+bn⇒an+bn-abn-1-an-1b=0⇒a·an-1-an-1b+b·bn-1-abn-1=0⇒an-1(a-b)-bn-1(a-b)=0⇒an-1-bn-1(a-b)=0⇒a-b=0⇒a=b[notpossible as a≡b]

Step 2: Find the value of n

an-1-bn-1=0⇒an-1=bn-1⇒an-1bn-1=1⇒abn-1=ab0⇒n-1=0⇒n=1

Hence, option C is correct.


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