Question

what is the 15th term of the geometric sequence -√5, √10, -2√10

Answer 1

do you know what a geometric sequence is?

Answer 2

yeah its a sequence with a common ratio

Answer 3

I just suck at radicals :(

Answer 4

got any ideas what the common ratio might be?

Answer 5

I got an equation an=a1 x r ^(n-1)

Answer 6

a1= first term, r= common ratio and n=term

Answer 7

anyhow the difference between "a term" and the "next term" is a multiplier

so if you just divide the "next term" by the "a term" then your quotient would be the multiplier or "common ratio" so let us use the 1st and 2nd term to get it \(\bf \cfrac{\sqrt{10}}{-\sqrt{5}}\implies \cfrac{\sqrt{2\cdot 5}}{-\sqrt{5}}\implies \cfrac{\sqrt{2}\cdot \cancel{\sqrt{5}}}{-\cancel{\sqrt{5}}}\) can you see the common ratio ?

Answer 8

is it √2 ?

Answer 9

well.... notice the - at the bottom, so it's negative

Answer 10

so \(\Large \bf a_{\color{red}{ n}}=a_1\cdot r^{{\color{red}{ n}}-1}
\\ \quad \\
a_1=\textit{first term in the sequence}\qquad r=\textit{common ratio}
\\ \quad \\
\textit{so the 15th term will be }\implies \Large a_{\color{red}{ 15}}=a_1\cdot r^{{\color{red}{ 15}}-1}\)

Answer 11

\(\Large \bf a_{\color{red}{ n}}=a_1\cdot r^{{\color{red}{ n}}-1}
\\ \quad \\
a_1=-\sqrt{5}\qquad r=-\sqrt{2}
\\ \quad \\
\textit{so the 15th term will be }\implies \Large a_{\color{red}{ 15}}=-\sqrt{5}\cdot (-\sqrt{2})^{{\color{red}{ 15}}-1} \)

Answer 12

so
-√5 X -√2^14

Answer 13

yeap

Answer 14

but I don't know how to solve (-√2)^14

Answer 15

hmm \(\bf a_{\color{red}{ 15}}=-\sqrt{5}\cdot (-\sqrt{2})^{{\color{red}{ 15}}-1}
\\ \quad \\
(-\sqrt{2})^{14}\implies (-1\cdot \sqrt{2})^{14}\implies (-1)^{14}\cdot (\sqrt{2})^{14}\implies 1\cdot \sqrt{2^{14}}\)

Answer 16

the answer choices are
1) -128√5
2) 128 √10
3) -16384 √5
4) 16384 √10

Answer 17

so what would you get from say \(\bf (-\sqrt{2})^{14}\implies (-1\cdot \sqrt{2})^{14}\implies (-1)^{14}\cdot (\sqrt{2})^{14}\implies 1\cdot \sqrt{2^{14}}\) ?

Answer 18

you would get √16384
and so how would you do how would you do √16384 x -√5

Answer 19

well.... you wouldn't

Answer 20

but you can get square root of 16384

Answer 21

so -128 √5 is the answer right

Answer 22

\(\bf (-\sqrt{2})^{14}\implies (-1\cdot \sqrt{2})^{14}\implies (-1)^{14}\cdot (\sqrt{2})^{14}\implies 1\cdot \sqrt{2^{14}}
\\ \quad \\
\sqrt{2^{14}}\implies \sqrt{(2^7)^2}\implies \Large \sqrt[{\color{red}{ 2}}]{(2^7)^{\color{red}{ 2}}}\implies 2^7\)

Answer 23

2^7 = 128
so 128 x -√5 = -128 √5 right ?

Answer 24

thanks for your help