SOLUTION: (a)0.5+0.25+0.125+.....+9.765625 using formula (i)

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Question 1208833: (a)0.5+0.25+0.125+.....+9.765625 using formula (i)
Found 2 solutions by ikleyn, AnlytcPhil:
Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.
(a)0.5+0.25+0.125+.....+9.765625 using formula (i)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Makes no sense.

This writing is nonsensical and anti-mathematical (= is written unprofessionally).

Please do not post nonsense to this forum.



Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!

I noticed that the decimal you gave 9.765625 when multiplied by 10-4
is indeed an approximation of one of the terms of the series, so maybe you left
off the " x 10-4 ".  I'll assume your problem was supposed to be:  

Find the sum of this series:



using the formula S%5Bn%5D%22%22=%22%22a%5B1%5D%28%281-r%5En%29%2F%281-r%5E%22%22%29%29

We know this is a geometric series with a%5Bn%5D=matrix%281%2C3%2C9.765625%2C%22x%22%2C10%5E%28-4%29%29
matrix%281%2C2%2Ccommon%2Cratio%29%22%22=%22%22r%22%22=%22%22matrix%281%2C2%2C2nd%2Cterm%29%2Fmatrix%281%2C2%2C1st%2Cterm%29%22%22=%22%220.25%2F0.5=0.5%22%22=%22%22matrix%281%2C2%2C3rd%2Cterm%29%2Fmatrix%281%2C2%2C2nd%2Cterm%29%22%22=%22%220.125%2F0.25%22%22=%22%220.5

But first we must find n, the number of terms, using the formula for the
nth term:

a%5Bn%5D%22%22=%22%22a%5B1%5Dr%5E%28n-1%29

matrix%281%2C3%2C9.765625%2C%22x%22%2C10%5E%28-4%29%29%22%22=%22%220.5%2A%280.5%29%5E%28n-1%29

matrix%281%2C3%2C9.765625%2C%22x%22%2C10%5E%28-4%29%29%22%22=%22%22%280.5%29%5En

We solve for n by taking the log base 10 of both sides

log%28%289.765625%29%29%2Blog%28%2810%5E%28-4%29%29%29%22%22=%22%22log%28%280.5%5En%29%29

log%28%289.765625%29%29-4%29%22%22=%22%22n%2Alog%28%280.5%29%29

0.9897000434-4%22%22=%22%22n%2A%28-0.3010299957%29

-3.010299957%2F%28-0.3010299957%29%22%22=%22%22n

9.999999999%22%22=%22%22n

We round that to 10

10%22%22=%22%22n

So there are n=10 terms in the series:

Using the formula:

S%5Bn%5D%22%22=%22%22a%5B1%5D%28%281-r%5En%29%2F%281-r%5E%22%22%29%29

S%5B10%5D%22%22=%22%220.5%28%281-0.5%5E10%29%2F%281-0.5%5E%22%22%29%29

S%5B10%5D%22%22=%22%220.5%28%280.9990234375%29%2F0.5%29%29

S%5B10%5D%22%22=%22%220.9990234375 <---ANSWER

So you can see that the sum of the series to 10 terms is 
getting close to 1.

Edwin