Question

Identify the 8th term of the given geometric sequence. 1/2, 2, 8, 32,...

Answer 1

Geometric sequence.

A geometric sequence goes from one term to the next by always multiplying or dividing by the constant value except 0. The constant number multiplied (or divided) at each stage of a geometric sequence is called the common ratio (r).

We have:

`a_1=(1)/(2)\\\\a_2=2\\\\a_3=8\\\\a_4=32\\\vdots`

Find the common ratio:

`\boxed{r=(a_n)/(a_(n-1))}\Rightarrow r=(2)/((1)/(2))=(8)/(2)=(32)/(8)=...\\\\\boxed{r=4}`

The formula of the n-th term of a geometric sequence:

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

Substitute:

`a_1=(1)/(2),\ r=4,\ n=8\\\\a_8=(1)/(2)\cdot4^(8-1)=(1)/(2)\cdot4^7=(1)/(2)\cdot16384\\\\\huge\boxed{a_8=8192}`

Method step by step:

`a_4=32,\ r=4\\\\a_5=a_4\cdot r\Rightarrow a_5=32\cdot4=128\\\\a_6=a_5\cdot r\Rightarrow a_6=128\cdot4=512\\\\a_7=a_6\cdot r\Rightarrow a_7=512\cdot4=2048\\\\a_8=a_7\cdot r\Rightarrow a_8=2048\cdot4=8192`