Week 2: Geometric Sequence

Published by on February 12, 2018 | Leave a response

In a Geometric Sequences, each term is found by multiplying the previous term by a constant called the “common ratio.”
In general, we could write an arithmetic sequence like this:
{t_1, t_1.r, t_1.r^2, t_1.r^3,…}
where:
t_1 is the first term
r is the “common ratio”

We can write a Geometric Sequence as a formula:
t_n = t_1.r^(n-1)

To sum up the terms of a Geometric Sequence:
t_1 + t_2 + t_3 +…
=t_1 + t_1.r + t_1.r^2 +…
Formula: S_n = \frac{t_1.(r^n-1)}{r-1}

Example: A construction company intends to build a six-storey tower. Each floor is three quarters of the previous floor, and the base of the tower is 512m^2. Provided that a tile used to pave the floors is 900cm^2, how many tiles would it take to pave the whole tower, at minimum?

Know:
t_1 = 512
r = 3/4 = 0.75
n = 6
The total area of the floors:
S_n = \frac{t_1.(r^n-1)}{r-1}
S_6 = \frac{512.(0.75^6-1)}{0.75-1}
S_n = 1683.55…(m^2) = 16835000 (cm^2)
The number of tiles needed at minimum:
\frac{S_n}{900} = 18705.56 tiles

Conclusion: Need 18706 tiles to pave the whole tower, at minimum

*Infinite Geometric Series
In an Infinite Geometric Series, when 0r>0, it will be a converging series
To calculate the sum, we use this formula: S_\infty = \frac{t_1}{1-r}

Example: If that company builds the tower to infinity, what is the total area of the floors?

Know:
t_1 = 512
r = 0.75
S_\infty = \frac{t_1}{1-r}
S_\infty = \frac{512}{1-0.75}
S_\infty = 2048 (m^2)

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