consider the following geometric sequence 400, 320, 256, 204.8, ......

i) what is the recurrance system that descibes this sequence?
( denote the sequence by xn, and its first term by x1.)

ii) find the closed form for this sequence.

iii) use the closed form from (ii) to find the tenth term of the sequence, giving your answer correct to 4 d.p.

consider the following geometric sequence 400, 320, 256, 204.8, ......

i) what is the recurrance system that descibes this sequence?
( denote the sequence by xn, and its first term by x1.)

ii) find the closed form for this sequence.

iii) use the closed form from (ii) to find the tenth term of the sequence, giving your answer correct to 4 d.p.
The common ratio is $\displaystyle \frac{400}{320} = \frac{5}{4}$.

Each term therefore is $\displaystyle x_{n+1} = x_nr = \frac{5}{4}x_n$.

So $\displaystyle x_2 = \frac{5}{4}x_1$

$\displaystyle x_3 = \frac{5}{4}x_2 = \frac{5}{4}\frac{5}{4}x_2 = \left(\frac{5}{4}\right)^2x_1$

$\displaystyle x_4 = \frac{5}{4}x_3 = \frac{5}{4}\left(\frac{5}{4}\right)^2 x_2 = \left(\frac{5}{4}\right)^3x_1$.

So in terms of $\displaystyle x_1$,

$\displaystyle x_{n+1}=\left(\frac{5}{4}\right)^n x_n$.

consider the following geometric sequence 400, 320, 256, 204.8, ......

i) what is the recurrance system that descibes this sequence?
( denote the sequence by xn, and its first term by x1.)

ii) find the closed form for this sequence.

iii) use the closed form from (ii) to find the tenth term of the sequence, giving your answer correct to 4 d.p.

Supposing that we had a series

$\displaystyle S_n = x_1 + x_1 r + x_1 r^2 + \dots x_1 r^n$.

Multiply this whole series by r, we get

$\displaystyle rS_n = x_1 r + x_1 r^2 + x_1 r^3 + \dots x_1 r^{n+1}$.

Subtract $\displaystyle S_n$ from $\displaystyle rS_n$ and we get

$\displaystyle rS_n - S_n = x_1 r^{n+1} - x_1$

$\displaystyle S_n(r - 1) = x_1 (r^{n+1} - 1)$

$\displaystyle S_n = \frac{x_1 (r^{n+1} - 1)}{r - 1}$.

So if $\displaystyle x_1 = 400$ and $\displaystyle r = \frac{5}{4}$ the closed form for this series is

$\displaystyle S_n = \frac{400 \left(\frac{5}{4}^{n+1} - 1\right)}{\frac{5}{4} - 1}$.

$\displaystyle S_n = \frac{400 \left(\frac{5}{4}^{n+1} - 1\right)}{\frac{1}{4}}$

$\displaystyle S_n = 1600 \left(\frac{5}{4}^{n+1} - 1\right)$.

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