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02 May 2007, 05:11
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What is the approximate value of the sum of this sequence:
S=1/32 + 1/33 + ... +1/64
Is there any shortcut?
S=1/32 + 1/33 + ... +1/64
Is there any shortcut?
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02 May 2007, 22:50
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You've got a Harmonic Progression when the reciprocals of the values of your series became an Arithmetic Progression. In your case the AP is
32, 33, 34,....64
The only way I know to solve these is to find a range within which the sum will be. Here's how:
Find the total number of terms in your series. Here's how I do it:
tn = t1 + (n-1)d
64 = 32 + (n-1)1
n = 33
Add the first term and the last term to itself n number of times (33 here)
Thats 33(1/32) & 33(1/64)
So the sum of the series will be between 33/64 & 33/32.
32, 33, 34,....64
The only way I know to solve these is to find a range within which the sum will be. Here's how:
Find the total number of terms in your series. Here's how I do it:
tn = t1 + (n-1)d
64 = 32 + (n-1)1
n = 33
Add the first term and the last term to itself n number of times (33 here)
Thats 33(1/32) & 33(1/64)
So the sum of the series will be between 33/64 & 33/32.
03 May 2007, 13:51
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i really wouldn't know how to solve this either, i'm curious to see the answer choices, but i would take this as a simple estimation problem
S=1/32 + 1/33 + ... +1/64
so you're adding a bunch of tiny fractions that are roughly equal to:
8/30 + 10/40 + 10/50 + 5/60
so you can add them rather quickly..
8/30 + 1/4 + 1/5 + 5/60 ->
4/15 + 1/4 + 3/15 + 5/60 ->
16/60 + 1/4 + 12/60 + 5/60 ->
33/60 + 1/4 ->
roughly 1/2 + 1/4 = 3/4
and since everything was rounded down i'd look for an answer a little higher. i would hope this could get me in the ballpark of the answers but you'd really have to see the answer choices to know for sure
.. and thats my very simple minded attempt to solve that =)
S=1/32 + 1/33 + ... +1/64
so you're adding a bunch of tiny fractions that are roughly equal to:
8/30 + 10/40 + 10/50 + 5/60
so you can add them rather quickly..
8/30 + 1/4 + 1/5 + 5/60 ->
4/15 + 1/4 + 3/15 + 5/60 ->
16/60 + 1/4 + 12/60 + 5/60 ->
33/60 + 1/4 ->
roughly 1/2 + 1/4 = 3/4
and since everything was rounded down i'd look for an answer a little higher. i would hope this could get me in the ballpark of the answers but you'd really have to see the answer choices to know for sure
.. and thats my very simple minded attempt to solve that =)
) is 0.7166.
