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tatiyna
4 years ago
7

Find S sub 12 for the geometric series: 1.5+ (-3) + 6 +....

Mathematics
1 answer:
Dominik [7]4 years ago
7 0

Answer:

S12 for geometric series: 1.5+ (-3) + 6 +.... would be: -2047.5

Step-by-step explanation:

Given the sequence to find the sum up-to 12 terms

1.5+ (-3) + 6 +....

A geometric sequence has a constant ratio 'r' and is defined by

a_n=a_1\cdot r^{n-1}

\mathrm{Compute\:the\:ratios\:of\:all\:the\:adjacent\:terms}:\q\:r=\frac{a_{n+1}}{a_n}

\frac{\left(-3\right)}{1.5}=-2,\:\quad \frac{\left(6\right)}{\left(-3\right)}=-2

\mathrm{The\:ratio\:of\:all\:the\:adjacent\:terms\:is\:the\:same\:and\:equal\:to}

r=-2

\mathrm{The\:first\:element\:of\:the\:sequence\:is}

a_1=1.5

as

a_n=a_1\cdot r^{n-1}

\mathrm{Therefore,\:the\:}n\mathrm{th\:term\:is\:computed\:by}\:

a_n=1.5\left(-2\right)^{n-1}

\mathrm{Geometric\:sequence\:sum\:formula:}

a_1\frac{1-r^n}{1-r}

\mathrm{Plug\:in\:the\:values:}

n=12,\:\spacea_1=1.5,\:\spacer=-2

   =1.5\cdot \frac{1-\left(-2\right)^{12}}{1-\left(-2\right)}

   =1.5\cdot \frac{1-\left(-2\right)^{12}}{1+2}

\mathrm{Multiply\:fractions}:\quad \:a\cdot \frac{b}{c}=\frac{a\:\cdot \:b}{c}

  =\frac{\left(1-\left(-2\right)^{12}\right)\cdot \:1.5}{1+2}

  =\frac{-6142.5}{1+2}                                          ∵ \left(1-\left(-2\right)^{12}\right)\cdot \:1.5=-6142.5

  =\frac{-6142.5}{3}

\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{-a}{b}=-\frac{a}{b}

  =-\frac{6142.5}{3}

  =-2047.5

Thus, S12 for geometric series: 1.5+ (-3) + 6 +.... would be: -2047.5

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