SOLUTION: Find the number of terms in this geometric series -4 + 16 - 64 + 256..., where Sn=52428

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Question 1193516: Find the number of terms in this geometric series
-4 + 16 - 64 + 256..., where Sn=52428

Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find the number of terms in this geometric series
-4 + 16 - 64 + 256..., where Sn=52428
~~~~~~~~~~~~~~~~ 

Use the formula for the sum of a geometric progression

    S%5Bn%5D = a%2A%28%28r%5En-1%29%2F%28r-1%29%29,


where "a" is the first term and "r" is the common ratio.


In your case a= -4, r= -4, so the formula is

    52428 = %28-4%29%2A%28%28%28-4%29%5En-1%29%2F%28-4-1%29%29,

or

    52428 = %28-4%29%2A%28%28%28-4%29%5En-1%29%2F%28-5%29%29.


From this formula,

    %28-4%29%5En+-+1 = 52428%2A%28%28-5%29%2F%28-4%29%29 = 65535

    %28-4%29%5En = 65535 + 1 = 65536 = 4%5E8,

so

    n = 8.


ANSWER.  n = 8.

Solved.

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On geometric progressions,  see introductory lessons
    - Geometric progressions
    - The proofs of the formulas for geometric progressions
    - Problems on geometric progressions
    - Word problems on geometric progressions
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic
"Geometric progressions".

Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


You should know and understand the formal algebraic solution shown by the other tutor.

But the answer can be obtained easily using logical reasoning and some easy approximations.

It should be clear by looking at the sums of the given numbers that the sum of the series is going to be negative whenever the number of terms is odd and positive whenever the number of terms is even.

Looking at the sums a bit more closely, it can be seen the the absolute value of the sum is always a bit less than the absolute value of the last term.

With a common ratio of -4 between terms, we know the term two to the right of a given term is going to be 16 times that term..

The 4th term (256) is even; since we have an even sum, the number of terms has to be even.

The 6th term is 16*256 which is roughly 4000, which is still smaller than the given sum of 52428.

The 8th term will be approximately 16 times 4000, or 64000; and the given sum IS a bit less than that.

So the number of terms has to be 8.