Question

What is the 10th term 1,
1/2,1/4

Answer 1

The 10th term will be:

`a_(10)=(1)/(512)`

Explanation:

Considering the sequence

`1,\:(1)/(2),\:(1)/(4)...`

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

`a_n=a_0\cdot r^(n-1)`

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

`((1)/(2))/(1)=(1)/(2),\:\quad ((1)/(4))/((1)/(2))=(1)/(2)`

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

`r=(1)/(2)`

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

`a_1=1`

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

`a_n=\left((1)/(2)\right)^(n-1)`

Putting n = 10 to determine in the nth term to determine the 10th term

`a_(10)=\left((1)/(2)\right)^(10-1)`

`a_(10)=\left((1)/(2)\right)^9`

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

`a_(10)=(1^9)/(2^9)`

`a_(10)=(1)/(2^9)` ∵
`1^9=1`

`a_(10)=(1)/(512)` ∵
`2^9=512`

Therefore, the 10th term will be:

`a_(10)=(1)/(512)`