Question

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

Answer 1

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