The sum of an infinite geometric sequence

We add more and more terms of the sequence $\displaystyle 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$:

• $\displaystyle {S_1}={1}$, $\displaystyle {S_2}={1.5}$, $\displaystyle {S_3}={1.75}$, $\displaystyle {S_4}={1.875}$, …
• Each new term is half of what is still missing up to 2, so the sum gets closer to 2 and never passes it.

On the graph – the sums $\displaystyle S_n$ and the line $\displaystyle {y}={2}$ (dashed):

2 is the sum of the infinite geometric sequence $\displaystyle 1, \frac{1}{2}, \frac{1}{4}, \ldots$

When $\displaystyle -1<q<1$, the power $\displaystyle q^n$ gets very small as $\displaystyle n$ grows: $\displaystyle \left(\frac{1}{2}\right)^{10}<0.001$. So $\displaystyle {S_n}={\frac{a_1(q^n-1)}{q-1}}$ approaches $\displaystyle \frac{a_1(0-1)}{q-1}$:

In the example: $\displaystyle {S}={\frac{1}{1-\frac{1}{2}}}={2}$.
When $\displaystyle q\geq1$ or $\displaystyle q\leq-1$ the terms do not get smaller, and the sequence has no such sum.

In the sequence $\displaystyle 12, -6, 3, -1.5, \ldots$, $\displaystyle {q}={-\frac{1}{2}}$:

The sums $\displaystyle 12, 6, 9, 7.5, \ldots$ jump above 8 and below it, each time closer to it.

In an infinite geometric sequence, the sum of the first and the fifth term is 5,440.
The first term is 8 times the fourth.

• $\displaystyle {\frac{a_1}{a_4}}={\frac{1}{q^3}}={8}$, so $\displaystyle {q^3}={\frac{1}{8}}$, and $\displaystyle {q}={\frac{1}{2}}$.
• $\displaystyle {a_1+a_1\cdot\frac{1}{16}}={5440}$, that is $\displaystyle {\frac{17}{16}a_1}={5440}$, and $\displaystyle {a_1}={5120}$.
• The sum of the sequence: $\displaystyle {S}={\frac{5120}{1-\frac{1}{2}}}={10240}$.
• The terms in the even places are an infinite geometric sequence with $\displaystyle {a_2}={2560}$ and ratio $\displaystyle {q^2}={\frac{1}{4}}$: their sum is $\displaystyle {\frac{2560}{1-\frac{1}{4}}}={\frac{10240}{3}}$.

In a triangle of area $\displaystyle S$ we draw its three midsegments. In the triangle they form we draw the midsegments again, and so on:

• The midsegments split a triangle into four congruent triangles, so the area of the middle triangle is $\displaystyle \frac{S}{4}$.
• The areas $\displaystyle S, \frac{S}{4}, \frac{S}{16}, \ldots$ are a geometric sequence, $\displaystyle {q}={\frac{1}{4}}$.
• The sum of all the areas: $\displaystyle {\frac{S}{1-\frac{1}{4}}}={\frac{4S}{3}}$.

Calculate the sum of the infinite geometric sequence.

The sum of an infinite geometric sequence is 60, and its ratio is $\displaystyle {q}={\frac{1}{3}}$.
Find $\displaystyle a_1$.

The sum of an infinite geometric sequence is 24, and its first term is 18.
Find $\displaystyle q$.

Which of the infinite geometric sequences have a sum? Explain.

• $\displaystyle 5, 10, 20, \ldots$
• $\displaystyle 5, -5, 5, \ldots$
• $\displaystyle 9, 6, 4, \ldots$

A ball falls from a height of 8 m.
After each bounce it rises to $\displaystyle \frac{3}{4}$ of the height before, and falls again.
How many metres will the ball travel, down and up, until it stops?