The sum of the first eight terms in a Geometric Series is 19680 and the sum of the first four terms is 240. A) Find the first term. B) Find

Question

The sum of the first eight terms in a Geometric Series is 19680 and the sum of the first four terms is 240. A) Find the first term. B) Find the common ratio. C) Justify your answers by showing steps that demonstrates your answers generate S8=19680 and S4=240.

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MichaelMet 3 years 2021-08-23T14:45:20+00:00 1 Answers 16 views 0

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    2021-08-23T14:46:30+00:00
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    Answer:

    First Term = 6

    Common Ratio = 3

    Step-by-step explanation:

    According to the Question,

    • Given, The sum of the first eight terms in a Geometric Series is 19680 and the sum of the first four terms is 240 .

    Thus, S_{8} = 19680 & S_{4} = 240 .

    • The Sum of n-term of Geometric Mean is S_{n} = \frac{a(r^{n-1)} }{r-1} Where, r>1 , a=First term of G.P & r=common Ratio .

    Now, on solving  \frac{S_{8} }{S_{4} }  we get,

    \frac{19680}{240} = \frac{\frac{a(r^{8-1)} }{r-1}}{\frac{a(r^{4-1)} }{r-1}}  

    82 = \frac{r^{8}-1 }{r^{4}-1 }

    82r^{4}-82 = r^{8}-1\\r^{8}-82r^{4}+81 = 0\\r^{8}-81r^{4}-r^{4}+81 = 0\\(r^{4}-81)( r^{4}-1) =0(r=1 is not possible so neglect ( r^{4}-1) =0 )

    So, r=3 Now Put this value in S_{4} = {\frac{a(r^{4-1)} }{r-1}} We get a=6 .

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