Week 2- My Geometric Sequences

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In the second week of Precalc, we have been taught about Geometric Sequences/Series and how to find the t_n , r(common ratio), S_n , S_\infty , and other equations. 

My Sequence is:

6, 18, 54…

Find the r:

r= \frac{18}{6}

r=3

Find the next 3 terms:

t_4 = 54×3

t_4 = 162

t_5 = 162×3

t_5 = 486

t_6 = 486×3

t_6= 1458

6, 18, 54, 162, 486, 1458…

Find t_{10} :

t_{10} = a r^{n-1}

t_{10} = 6 ( 3^9 )

t_{10} = 6 (19683)

t_{10} = 118098

Find S_{10} :

S_{10} = a ( r^n - 1) \div \ (r-1)

S_{10} = 6 ( 3^{10} - 1) \div\ 3-1

S_{10} = 6 (59048) \div \ 2

S_{10} = \frac {354388}{2}

S_{10} = 177144

Does this Sequence converge or diverge?

This Sequence converges because the number of the terms increase by the common ratio of 3. If any sequence were to diverge the terms would decrease and you would be able to find S_\infty by using the fomula: S_\infty = a \div \ 1-r

 

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