# Thread: Geometric sequence & series

1. ## Geometric sequence & series

A sequence of numbers u_1, u_2, …, u_n, … is given by the formula $\displaystyle u_n = 3(\frac{2}{3})^n - 1$ where n is a positive integer.
(a) Find the value of u_1, u_2 and u_3.
(b) Show that $\displaystyle \sum_{n=1}^{15} u_n = -9.014$ to 4 s.f.
(c) Proved that $\displaystyle u_{n+1} = 2 (\frac{2}{3})^3 - 1$.
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(a) u_1 = 1, u_2 = 1/3 & u_3 = -1/9.

How could I do rest of them?

2. Originally Posted by geton
A sequence of numbers u_1, u_2, …, u_n, … is given by the formula $\displaystyle u_n = 3(\frac{2}{3})^n - 1$ where n is a positive integer.
(a) Find the value of u_1, u_2 and u_3.
(b) Show that $\displaystyle \sum_{n=1}^{15} u_n = -9.014$ to 4 s.f.
(c) Proved that $\displaystyle u_{n+1} = 2 (\frac{2}{3})^3 - 1$.
------------------------

(a) u_1 = 1, u_2 = 1/3 & u_3 = -1/9.

How could I do rest of them?
The sum of n terms in a geometric series

S_n= (a+k)+(at+k)+(at^2+k)+(at^3+k)....+(at^n+k)

ie; series with first term a and common factor t is
$\displaystyle S_n =\frac{a(t^n-1)}{t-1} + (n+1)k$

you have $\displaystyle a= u_1 = 1$
$\displaystyle t=\frac{2}{3}$

k= (-1) and n =15

put them in the formula

about 3rd question I think its incorrect