Need help with finding Nth formula for a sequence

[sufi]

Hi can anyone help me to get the nth value of this sequence given the recursive definition?

Find, showing all working, a formula for the \(\displaystyle n\)-th term \(\displaystyle t_n\) of the sequence \(\displaystyle \left(t_n\right)\) defined by:

. . . . .\(\displaystyle t_1\, =\, 6;\, t_n\, =\, \dfrac{-3t_{n-1}}{2},\, n\, \geq\, 2\)

if you can help me understand how to solve these it will be cool

 

[stapel]

Hi can anyone help me to get the nth value of this sequence given the recursive definition?

Find, showing all working, a formula for the \(\displaystyle n\)-th term \(\displaystyle t_n\) of the sequence \(\displaystyle \left(t_n\right)\) defined by:

. . . . .\(\displaystyle t_1\, =\, 6;\, t_n\, =\, \dfrac{-3t_{n-1}}{2},\, n\, \geq\, 2\)

if you can help me understand how to solve these it will be cool

"The n-th value" is "the closed form" (that is, the nice, neat, formulaic format). So you need to find the "closed form" for this recursive function.

Thank you for indicating that you're not familiar with the methods for this topic. To get you started, you might find this online lesson to be helpful.

 

[Deleted member 4993]

Hi can anyone help me to get the nth value of this sequence given the recursive definition?


if you can help me understand how to solve these it will be cool

By observation, you can see that it is a geometric series with a ratio r = -3/2.

What is the closed form expression for the nth term in a geometric series?

 

[HallsofIvy]

Subhotosh Kahn, there is nothing said here about a "series". sufi just says the "nth term of this sequence".

sufi, look at a few terms: \(\displaystyle t_1= 6\), \(\displaystyle t_2= (-3/2)t_1= 6(-3/2)\), \(\displaystyle t_2= (-3/2)t_1= 6(-3/2)^2\), \(\displaystyle t_3= (-3/2)t_2= 6(-3/2)^3\).

You are simply multiplying each time by -3/2 so you get powers of -3/2.

 

[Deleted member 4993]

Subhotosh Kahn, there is nothing said here about a "series". sufi just says the "nth term of this sequence".

sufi, look at a few terms: \(\displaystyle t_1= 6\), \(\displaystyle t_2= (-3/2)t_1= 6(-3/2)\), \(\displaystyle t_2= (-3/2)t_1= 6(-3/2)^2\), \(\displaystyle t_3= (-3/2)t_2= 6(-3/2)^3\).

You are simply multiplying each time by -3/2 so you get powers of -3/2.

Yes - I made a blunder of using wrong name for "sequence".