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Could someone give me a hint on how I could do this question( it is a non- calculator question):

The 5th term of a geometric series is 12 and the 7th term is 3. Find the two possible values of the sum to infinity of the series

Maths2468
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2 Answers2

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If $a,r$ be the first term and the common ratio respectively,

$ar^{5-1}=12, ar^{7-1}=3\implies r^2=\dfrac14$

The infinite sum $=\dfrac a{1-r}$

lab bhattacharjee
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  • But its a geometric series right. Dont we have to equate the 12 to u1(r^5-1)/(r-1)? – Maths2468 Apr 17 '16 at 11:14
  • @Maths2468, See https://en.wikipedia.org/wiki/Geometric_progression#Elementary_properties and https://en.wikipedia.org/wiki/Geometric_progression#Infinite_geometric_series – lab bhattacharjee Apr 17 '16 at 11:18
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Let the first term be $a$ and the common ratio be $r$.

Then, $$ar^4=12$$ $$ar^6=3$$ Dividing the two, $$r^2\frac14$$ $$r=\pm\frac12$$

Using this, the sum can be found.

GoodDeeds
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  • But its a geometric series right. Dont we have to equate the 12 to u1(r^5-1)/(r-1)? – Maths2468 Apr 17 '16 at 11:13
  • @Maths2468 That is the sum of the first five terms of the geometric series. The fifth term of the series will be the term of the corresponding sequence. – GoodDeeds Apr 17 '16 at 11:15
  • oh that makes the question far far easier! I thought the word series was only used when a sequence was being added. – Maths2468 Apr 17 '16 at 11:18
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    @Maths2468: Yes, "series" generally implies the sum of a sequence, but not always. IMHO, the wording of that question is sloppy for using the word "series" like that. It should say "geometric sequence" or "geometric progression". Feel free to deduct half a mark from your teacher. :) – PM 2Ring Apr 17 '16 at 11:57
  • @Maths2468 If not for the sequence, "term" of a series would make little sense. A series is a formal sum of its terms. What you referred to was the 5th partial sum of the series. – GoodDeeds Apr 17 '16 at 12:03