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Of how many terms is $ \dfrac{{55}}{{72}} $ , the sum of series $ \dfrac{2}{9}, - \dfrac{1}{3},\dfrac{1}{2}, - ... $ ?

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Last updated date: 14th Jul 2024
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Answer
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Hint: First we have to define what the terms we need to solve the problem are.
Since the given question is the form like in G.P terms are, \[a,(ar),(a{r^2}),(a{r^3}),...\] where $ a $ is the first term and $ r $ is the common ratio.
Formula used:
The sum of the terms is $ \dfrac{{a({r^n} - 1)}}{{r - 1}} $

Complete step by step answer:
Since the question is in the GP form so the first value is the $ a = \dfrac{2}{9} $ and now we will need to find the common ratio of the GP which is $ r = \dfrac{{\dfrac{{ - 1}}{3}}}{{\dfrac{2}{9}}} $ (second terms divides the first terms to find the common ratio)
Which gives as $ r = \dfrac{{ - 3}}{2} $ (is the common ratio)
Since we have three terms now, we are going to find four terms in this GP.
To find the fourth term in what terms divides the third term will gives as the $ r = \dfrac{{ - 3}}{2} $
That means $ \dfrac{?}{{\dfrac{1}{2}}} = \dfrac{{ - 3}}{2} $ this can be obtain only if the fourth term (clue is negative term as they given in the question) is $ \dfrac{{ - 3}}{4} $ which is $ \dfrac{{\dfrac{{ - 3}}{4}}}{{\dfrac{1}{2}}} = \dfrac{{ - 3}}{2} $
Hence the fourth term is $ \dfrac{{ - 3}}{4} $ ; similarly, we need to find the fifth term that is divides the fourth term needs to give the $ r = \dfrac{{ - 3}}{2} $ common ratio; which means the only term (it will be a positive term as the sequences of the terms changes positive, negative again and again) is $ \dfrac{{\dfrac{9}{8}}}{{\dfrac{{ - 3}}{4}}} = \dfrac{{ - 3}}{2} $ common ratio.
Finally, we have the five terms; which are $ \dfrac{2}{9}, - \dfrac{1}{3},\dfrac{1}{2},\dfrac{{ - 3}}{4},\dfrac{9}{8} $ now the sum of these terms will give the resultant which is $ \dfrac{2}{9} - \dfrac{1}{3} + \dfrac{1}{2} - \dfrac{3}{4} + \dfrac{9}{8} = \dfrac{{55}}{{72}} $
Hence there are five (5) numbers of terms needed to sum and get the required resultant.

Note: We can also able to solve this problem using $ \dfrac{{a({r^n} - 1)}}{{r - 1}} $ where $ a $ is the first term and $ r $ is the common ratio.
Since sum of the terms is given, substitute that all and finding the n be well the answer as the same five.