The value of k so that the terms forms a geometric sequence is
\begin{gathered}\rm 1.~k=\dfrac{1}{3}\\\\2.~k=5\end{gathered}
1. k=
3
1
2. k=5
Further explanation
Geometric sequences are series of numbers that have a constant ratio
or can be interpreted:
Each number is obtained by multiplying the previous number by a constant
The sequence can be:
\rm a,ar,ar^2,ar^3,ar^4 ... etc.a,ar,ar
2
,ar
3
,ar
4
...etc.
Can be formulated for xn
\boxed{\bold{x_n=ar^{(n-1)}}}
x
n
=ar
(n−1)
where:
a is the first term, and
r is common ratio
There are 3 ways to solve quadratic equations
factoring
perfect squared
quadratic formula
The terms k-3,k+1,and 4k-2 form a geometric sequence if
\rm \dfrac{k+1}{k-3}=\dfrac{4k-2}{k+1}
k−3
k+1
=
k+1
4k−2
we multiply in cross so that we will get the quadratic equation
\begin{gathered}\rm (k-3)(4k-2)=(k+1)(k+1)\\\\4k^2-14k+6=k^2+2k+1\\\\3k^2-16k+5=0\end{gathered}
(k−3)(4k−2)=(k+1)(k+1)
4k
2
−14k+6=k
2
+2k+1
3k
2
−16k+5=0
We solve this equation by factoring
For ax²+bx+c = ax²+ px+qx+c
pq = ac and p+q = b
So :
\begin{gathered}\rm 3k^2-16k+5=0\\\\3k^2-15k-k+5=0\Rightarrow p=-15,\:q=-1,\:a=3,\:c=5\\\\3k(k-5)-1(k-5)=0\\\\(3k-1)(k-5)=0\\\\3k-1=0\Rightarrow k=\boxed{\bold{\dfrac{1}{3}}}\\\\k-5=0\Rightarrow k=\boxed{\bold{5}}\end{gathered}
3k
2
−16k+5=0
3k
2
−15k−k+5=0⇒p=−15,q=−1,a=3,c=5
3k(k−5)−1(k−5)=0
(3k−1)(k−5)=0
3k−1=0⇒k=
3
1
k−5=0⇒k=
5
Learn more
the geometric sequence
https://brainly.ph/question/154960
Formula for geometric sequence
https://brainly.ph/question/2238923
Geometric
#BetterWithBrainly
hope it's help
#carry on learning buddy
