Question

Find the sum of the first 25 terms in this geometric series:
8 + 6 + 4.5...

Answer 1

it is 31.98 i did the test

Explanation:

Answer 2

Explanation:

Given the geometric sequence

8 + 6 + 4.5...

A geometric sequence has a constant ratio and is defined by

`a_n=a_1\cdot r^(n-1)`

`\mathrm{Compute\:the\:ratios\:of\:all\:the\:adjacent\:terms}:\quad \:r=(a_(n+1))/(a_n)`

`(6)/(8)=(3)/(4),\:\quad (4.5)/(6)=(3)/(4)`

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

`r=(3)/(4)`

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

`a_1=8`

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

`a_n=8\left((3)/(4)\right)^(n-1)`

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

`a_1(1-r^n)/(1-r)`

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

`n=25,\:\spacea_1=8,\:\spacer=(3)/(4)`

`=8\cdot (1-\left((3)/(4)\right)^(25))/(1-(3)/(4))`

`\mathrm{Multiply\:fractions}:\quad \:a\cdot (b)/(c)=(a\:\cdot \:b)/(c)`

`=(\left(1-\left((3)/(4)\right)^(25)\right)\cdot \:8)/(1-(3)/(4))`

`=(8\left(-\left((3)/(4)\right)^(25)+1\right))/((1)/(4))`

`\mathrm{Apply\:exponent\:rule}:\quad \left((a)/(b)\right)^c=(a^c)/(b^c)`

`=(8\left(-(3^(25))/(4^(25))+1\right))/((1)/(4))`

`\mathrm{Apply\:the\:fraction\:rule}:\quad (a)/((b)/(c))=(a\cdot \:c)/(b)`

`=(\left(1-(3^(25))/(4^(25))\right)\cdot \:8\cdot \:4)/(1)`

`\mathrm{Multiply\:the\:numbers:}\:8\cdot \:4=32`

`=(32\left(-(3^(25))/(4^(25))+1\right))/(1)`

`=(32\cdot (4^(25)-3^(25))/(4^(25)))/(1)` ∵
`\mathrm{Join}\:1-(3^(25))/(4^(25)):\quad (4^(25)-3^(25))/(4^(25))`

`=32\cdot (4^(25)-3^(25))/(4^(25))`

`=(\left(4^(25)-3^(25)\right)\cdot \:32)/(4^(25))`

`=(2^5\left(4^(25)-3^(25)\right))/(2^(50))` ∵
`\mathrm{Factor}\:32:\ 2^5`,
`\mathrm{Factor}\:4^(25):\ 2^(50)`

so

`=(4^(25)-3^(25))/(2^(45))` ∵
`\mathrm{Cancel\:}(\left(4^(25)-3^(25)\right)\cdot \:2^5)/(2^(50)):\quad (4^(25)-3^(25))/(2^(45))`

`\mathrm{Apply\:the\:fraction\:rule}:\quad (a\pm \:b)/(c)=(a)/(c)\pm (b)/(c)`

`=(4^(25))/(2^(45))-(3^(25))/(2^(45))`

`=32-(3^(25))/(2^(45))` ∵
`(4^(25))/(2^(45))=32`

`=32-0.024` ∵
`(3^(25))/(2^(45))=0.024`

`=31.98`

Therefore, the sum of the first 25 terms in this geometric series: 31.98