Not hard, but...

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Naheed
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Not hard, but...

Unread post by Naheed » Sat Feb 08, 2014 7:06 pm

We have a sequence of 100 positive integers.
The average of 1st and 2nd term is 1.
The average of 2nd and 3rd term is 2.
The average of 3rd and 4th term is 3. The pattern continues.

The average of 99th and 100th term is 99. Find the 100th term.

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Labib
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Re: Not hard, but...

Unread post by Labib » Sat Feb 08, 2014 7:40 pm

Here are some hints.
Let's define the $i$-th element of the sequence as $a_i$.
Hint $1$:
So, $\frac {a_1+a_2}2 = 1 \Rightarrow a_1+a_2 = 2$
and, $a_1,a_2 \geq 1$. Can you deduce the value of $a_0$ now?
Hint $2$:
Can you prove that, $a_i = a_{i-2} + 2$ for $i > 2$?
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Raiyan Jamil
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Re: Not hard, but...

Unread post by Raiyan Jamil » Tue Feb 11, 2014 7:22 pm

I think the ans is 100 . I'm confused . Please someone tell me the correct ans .
Last edited by Raiyan Jamil on Wed Feb 12, 2014 10:28 am, edited 1 time in total.
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Labib
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Re: Not hard, but...

Unread post by Labib » Tue Feb 11, 2014 11:15 pm

Labib wrote: Hint $1$:
So, $\frac {a_1+a_2}2 = 1 \Rightarrow a_1+a_2 = 2$
and, $a_1,a_2 \geq 1$. Can you deduce the value of $a_0$ now?
There was a typo in Hint $1$.Here's a corrected version.
Hint $1$:
So, $\frac {a_1+a_2}2 = 1 \Rightarrow a_1+a_2 = 2$
and, $a_1,a_2 \geq 1$. Can you deduce the value of $a_1$ and $a_2$ now?
And if you want to know the answer,
here it is =>
$99$
And please post how you solved the problem. It not only helps others, but also helps you find any possible bug in your solution.
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Raiyan Jamil
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Re: Not hard, but...

Unread post by Raiyan Jamil » Wed Feb 12, 2014 10:32 am

#Labib , how 99 ? I tried to make the sequence = 0 , 2 , 2 , 4 , 4 , 6 , 6 , 8 , 8 , 10 , 10 , .................... Like this I think the ans will be 100 . If not , it will be 98 . But 99 is a odd number and can't be contained in the sequence . So , how 99 ?
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*Mahi*
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Re: Not hard, but...

Unread post by *Mahi* » Wed Feb 12, 2014 11:09 am

Positive integers start from $1$, not $0$.
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Raiyan Jamil
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Re: Not hard, but...

Unread post by Raiyan Jamil » Wed Feb 12, 2014 11:36 am

Sorry I forgot . So the sequence will be 1 , 1 , 3 , 3 , 5 , 5 , 7 , 7 , 9 , 9 , 11 , 11................................ And the 100th term will be 99 .
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Sidharth
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Re: Not hard, but...

Unread post by Sidharth » Mon Feb 17, 2014 11:40 pm

Oooopppsss.....!sorry!I forgot that it must have to be an integer.My mistake........

Naheed
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Re: Not hard, but...

Unread post by Naheed » Mon Jan 05, 2015 9:37 pm

Thanks to all! The sequence is 1, 1, 3, 3,...and the 100th term is 99.

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