Can you solve it?
Find the sum up to 20 terms.
1 + 3 + 7 + 1 5 + 3 1 + … + T 2 0 ,
where ∖(T_n = 2^n- 1∖).
The answer is 2097130.
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2 solutions
I do not feel it is special series. As solved by Ikkyu San , it is only sum of two series.
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You're right
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any series which is not in A.P,G.P,A.G.P or H.P is considered as special series i thought like this may be i am wrong but may intention was to provide others method to find general term for this question
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@Raj Rajput – What ..... ?
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@Sai Ram – in this problem one can find out "general term" and i just posted a way to find that general term
@Calvin Lin SIR,WHY DID YOU GIVE THE GENERAL TERM?
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Please refrain from typing in all capital letters, as that is considered extremely rude on the internet.
For pattern recognition problems, we tend to prefer to be explicit with the terms, since there could be numerous other interpretations of what the pattern would be, which would then result in a different answer.
For example, you could be referencing the number of regions of a circle with n-1 cuts, or the Pentanacci numbers.
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I'm extremely sorry sir.I want to convey that giving the general term solves half of the problem.
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@Sai Ram – Without giving the general term, this problem is unsolvable. I can delete this problem if that is your preference.
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@Calvin Lin – The solvers will find out the general term as given in the first solution.
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@Sai Ram – Why must that be the only pattern?
We could also have T n = ∑ k = 1 i ( k n ) for any i ≥ 5 . Then, we would get a different answer each time.
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@Calvin Lin – I didn't understand what you actually mean.
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@Sai Ram – What he means is that defining a series by it's first 5 terms (or 10 terms, or 1 million terms) does not provide an unambiguous definition of the series for all terms. It might be likely that many readers to one particular definition, but there it is entirely possible to have multiple definitions of the series which still gives the the term : There is very rarely such a thing as an obvious definition. For instance if you did not give any particular definition for T n the set of terms could be defined as T n = 2 n − 1 : n < = 5 , T n = 0 : n > 5 - that definition would still be consistent with the problem as originally given.
Did it in the same way.
1 + 3 + 7 + 1 5 + 3 1 + ⋯ (up to 2 0 terms) can be represented by:
( 2 1 − 1 ) + ( 2 2 − 1 ) + ( 2 3 − 1 ) + ( 2 4 − 1 ) + ( 2 5 − 1 ) + ⋯ + ( 2 2 0 − 1 ) = n = 1 ∑ 2 0 ( 2 n − 1 ) = n = 1 ∑ 2 0 2 n − n = 1 ∑ 2 0 1 = 2 − 1 2 ( 2 2 0 − 1 ) − 2 0 ( 1 ) = 2 2 1 − 2 − 2 0 = 2 0 9 7 1 5 2 − 2 2 = 2 0 9 7 1 3 0
Really nice solution did the same way
Used the same method.
Same method