Series and sequences help (1 Viewer)

mkv_86 · Oct 29, 2017

The sum of the first n terms of a series is given by Sn= (x+1)^2n - 1. Show that the sequence is geometric.

Much thanks

jazz519 · Oct 29, 2017

mkv_86 said:
The sum of the first n terms of a series is given by Sn= (x+1)^2n - 1. Show that the sequence is geometric.

Much thanks

Just checking the 2n and -1 are separate yeah (as in the -1 is not to the power as well)

jazz519 · Oct 29, 2017

https://imgur.com/gallery/ofbau

(not really sure how else you could do it since there is no values given)

mkv_86 · Oct 29, 2017

jazz519 said:
https://imgur.com/gallery/ofbau

(not really sure how else you could do it since there is no values given)

looks about right, thank you!!!

1729 · Oct 29, 2017

mkv_86 said:
The sum of the first n terms of a series is given by Sn= (x+1)^2n - 1. Show that the sequence is geometric.

Much thanks

$$\noindent \begin{align*}\quad S_1 &= (x+1)^2-1 \\ \implies T_1 &= (x+1)^2 - 1 \\\\ S_2 &= (x+1)^4 - 1 = T_1 + T_2 \\ \implies T_2 &= S_2 - T_1 \\ &= (x+1)^4 - 1 - (x+1)^2 + 1 \\ &= (x+1)^2\left((x+1)^2 - 1\right) \\\\ S_3 &= (x+1)^6 - 1 = T_1 + T_2 + T_3 \\ \implies T_3 &= S_3 - S_2 \\ &= (x+1)^6-1 - (x+1)^4 + 1 \\ &= (x+1)^4\left((x+1)^2-1\right) \end{align*} \\ As seen, the sequence is geometric with $a = (x+1)^2 - 1$ and $r = (x+1)^2$

Series and sequences help (1 Viewer)

mkv_86

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