Question

Given the geometric sequence -1, 1/2, -1/4, 1/8, -1/16. What is the common ratio?

Answer 1

The given sequence is :

`-1,\text{ }(1)/(2),\text{ }(-1)/(4),\text{ }(1)/(8),\text{ }(-1)/(16)`

Geometric Sequence : A geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

The expression is :

`a,ar^1,ar^2\ldots\ldots.ar^n`

Common Ratio : A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence

In the given sequence :

First term = (-1)

Second Term = (1/2)

Ratio of the second term to the first term :

`\begin{gathered} \text{Common Ratio = }\frac{Second\text{ Term}}{First\text{ Term}} \\ \text{ Common Ratio=}((1)/(2))/((-1)) \\ \text{ Common Ratio=-}(1)/(2) \\ \text{ Common Ratio=}(-0.5) \end{gathered}`

Common Ratio = (-0.5)

The expression for the n-terms of Geometric sequence is :

`a_n=a_1r^(n-1)`

In the given system , first term = (-1) and the ratio r = (-1/2)

for 11 term, put n = 11

Substitute the value and simplify:

`\begin{gathered} a_n=a_1r^(n-1) \\ a_(11)=(-1)(-(1)/(2))^(11) \\ \text{ Since 11 is the odd number so, (-1)}^(odd)=\text{ (-1)} \\ a_(11)=(-1)(-(1)/(2^(11))) \\ \text{ }a_(11)=(-1)((-1)/(2048)^{}) \\ \text{ Since (-1) }*(-1)\text{ = 1} \\ a_(11)=(1)/(2048) \end{gathered}`

So, The 11 term is 1/2048

Answer : Common Ratio = (-0.5), 11 term is 1/2048