## What is Geometric Sequence

### Geometric Sequences

Geometric sequences are those in which the following number is obtained by multiplying the previous term by a constant known as the common ratio. The letter $$r$$ is used to show the common ratio.

We have an increasing geometric sequence if $$r$$ is more than $$1$$. We have a decreasing geometric sequence if $$r$$ is between $$0$$ and $$1$$.

### The Geometric Sequences Formula

We can use the geometric sequence formula to find any number in the sequence.

$$a_n = a \ r^{(n-1)}$$, where:

• $$a$$: The first term
• $$a_n$$: The $$n$$th term
• $$n$$: The term poistion
• $$r$$: Common ratio

### Finding the terms in a Geometric Sequence

If we divide any term by the previous term, we can find the common ratio: $$r \ = \ \frac{a_n}{a_{n \ - 1}}$$

### Example

Consider the geometric sequence: $$2, \ 6, \ 18, \ 54, \ 162, \ ...$$, Find $$a_{7} \ = \ ?$$

Solution

Before using the geometric sequence formula, we need to know the first term, the common ratio, and the position of the term we want to find:

• The first term: $$a \ = \ 2$$
• common ratio: $$r \ = \ 3$$
• Term's position: $$n \ = \ 7$$

Now, we fill in the formula with these numbers:

$$a_n = a \ r^{(n-1)} \ ⇒ \ a_7 = 2 \ 3^{(6)} \ = \ 2 \times 729 \ = \ 1458$$

We can see that the number is very big. Depending on what the common ratio is, geometric sequences tend to grow quickly.

### The formula for the Sum of a Geometric Sequence

With the sum of a finite geometric sequence formula, you can find the sum of a geometric sequence's first $$n$$ terms. Consider a geometric series with $$n$$ terms, where the first term is $$a$$, and the ratio $$r$$: $$a, \ ar, \ ar^2, \ ar^3, \ ... \ , \ ar^{n \ - \ 1}$$

The sum is shown by $$S_n$$ and given by the formula:

• $$S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1}$$, when $$r \ ≠ \ 1$$
• $$S_n \ = \ na$$, when $$r \ = \ 1$$

### Example

Add up the first four terms in: $$10, \ 30, \ 90, \ 270, \ 810, \ 2430, \ ...$$

Solution

• The first term: $$a \ = \ 10$$
• common ratio: $$r \ = \ 3$$
• $$n \ = \ 4$$

So:

$$S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ = \ \frac{10(3^n \ - \ 1)}{3 \ - \ 1} \ = \ 400$$

### Exercises for Geometric Sequences

1) Find the explicit formula: $$-1, \ 2, \ -4, \ 8, \ -16, \ ...$$

2) Find the explicit formula: $$-2, \ -6, \ -18, \ -54, \ -162, \ ...$$

3) Find the explicit formula and $$a_{11}$$: $$88, \ 44, \ 22, \ 11, \ 5.5, \ ...$$

4) Find the explicit formula and $$a_5$$: $$a \ = \ 5, \ r \ = \ -5$$

5) Find the explicit formula and the first $$7$$ terms: $$a \ = \ 2, \ r \ = \ \frac{1}{3}$$

6) Find the first $$10$$ terms: $$a \ = \ 7, \ r \ = \ -\frac{1}{2}$$

7) Find $$a_{7}$$: $$45, \ , \ 9, \ \frac{9}{5} \ , \ \frac{9}{25} \ , \ ...$$

8) Find $$a_{10}$$: $$a_7 \ = \ \frac{2}{27} \ , \ r \ = \ -\frac{1}{3}$$

9) Find the explicit formula and the sum of the first five terms: $$a \ = \ 10, \ r \ = \ 2$$

10) Find the explicit formula and the sum of the first $$10$$ terms: $$96, \ 48, \ 24, \ 12, \ 6, \ ...$$

1) Find the explicit formula: $$-1, \ 2, \ -4, \ 8, \ -16, \ ...$$

$$\color{red}{a \ = \ -1, \ r \ = \ \frac{2}{-1} \ = \ -2, \ a_n = a \ r^{(n-1)}}$$
$$\color{red}{⇒ \ a_n = -1 \ (-2)^{(n-1)}}$$

2) Find the explicit formula: $$-2, \ -6, \ -18, \ -54, \ -162, \ ...$$

$$\color{red}{a \ = \ -2, \ r \ = \ \frac{-6}{-2} \ = \ 3, \ a_n = a \ r^{(n-1)}}$$
$$\color{red}{⇒ \ a_n = -2 \ (3)^{(n-1)}}$$

3) Find the explicit formula and $$a_{11}$$: $$88, \ 44, \ 22, \ 11, \ 5.5, \ ...$$

$$\color{red}{a \ = \ 88, \ r \ = \ \frac{44}{88} \ = \ \frac{1}{2}, \ a_n = a \ r^{(n-1)}}$$
$$\color{red}{⇒ \ a_n = 88 \ (\frac{1}{2} \ )^{(n-1)} \ ⇒ \ a_{11} = 88 \ (\frac{1}{2} \ )^{(11-1)} \ = \ \frac{88}{1024} \ = \ \frac{11}{256}}$$

4) Find the explicit formula and $$a_5$$: $$a \ = \ 5, \ r \ = \ -5$$

$$\color{red}{a \ = \ 5, \ r \ = \ -5, \ a_n = a \ r^{(n-1)}}$$
$$\color{red}{⇒ \ a_n = 5 \ (-5)^{(n-1)} \ ⇒ \ a_5 = 5 \ (-5)^{(5-1)} \ = \ 5 \times \frac{1}{(-5)^4} \ = \ \frac{1}{125}}$$

5) Find the explicit formula and the first $$7$$ terms: $$a \ = \ 2, \ r \ = \ \frac{1}{3}$$

$$\color{red}{a \ = \ 2, \ r \ = \ \frac{1}{3} \ , \ a_n = a \ r^{(n-1)}}$$
$$\color{red}{⇒ \ a_n = 2 \ (\frac{1}{3} \ )^{(n-1)}}$$
$$\color{red}{⇒ 2, \ \frac{2}{3} \ , \ \frac{2}{9} \ , \ \frac{2}{27} \ , \ \frac{2}{81} \ , \ \frac{2}{243} \ , \ \frac{2}{729} \ , \ ...}$$

6) Find the first $$10$$ terms: $$a \ = \ 7, \ r \ = \ -\frac{1}{2}$$

$$\color{red}{a \ = \ 7, \ r \ = \ -\frac{1}{2} \ , \ a_n = a \ r^{(n-1)}}$$
$$\color{red}{⇒ \ a_n = 7 \ (-\frac{1}{2} \ )^{(n-1)}}$$
$$\color{red}{⇒ 7, \ -\frac{7}{2} \ , \ \frac{7}{4} \ , \ -\frac{7}{8} \ , \ \frac{7}{16} \ , \ -\frac{7}{32} \ , \ \frac{7}{64} \ , \ -\frac{7}{128} \ , \ \frac{7}{256} \ , \ -\frac{7}{512} \ , \ ...}$$

7) Find $$a_{7}$$: $$45, \ , \ 9, \ \frac{9}{5} \ , \ \frac{9}{25} \ , \ ...$$

$$\color{red}{a \ = \ 45, \ r \ = \ \frac{1}{5}, \ a_n = a \ r^{(n-1)}}$$
$$\color{red}{⇒ \ a_n = 45 \ (\frac{1}{5} \ )^{(n-1)} \ ⇒ \ a_7 = 45 \ (\frac{1}{5} \ )^{(7-1)} \ = \ 45 \times \frac{1}{5^6} \ = \ \frac{9}{3125}}$$

8) Find $$a_{10}$$: $$a_7 \ = \ \frac{2}{27} \ , \ r \ = \ -\frac{1}{3}$$

$$\color{red}{a_7 \ = \ \frac{2}{27} \ , \ r \ = \ -\frac{1}{3} \ , \ a_n = a \ r^{(n-1)}}$$
$$\color{red}{⇒ \ a_7 \ = \ a \ (-\frac{1}{3} \ )^{(7-1)} \ ⇒ \ a \ = \ \frac{a_7}{(-\frac{1}{3} \ )^6} \ = \ \frac{\frac{2}{27}}{\frac{1}{3^6}} \ = \ 54}$$
$$\color{red}{⇒ \ a_{10} = \ 54 \ (-\frac{1}{3} \ )^{(10-1)} \ = \ -\frac{2}{729}}$$

9) Find the explicit formula and the sum of the first five terms: $$a \ = \ 10, \ r \ = \ 2$$

$$\color{red}{a \ = \ 10, \ r \ = \ 2, \ a_n = a \ r^{(n-1)}}$$
$$\color{red}{⇒ \ a_n = 10 \ 2^{(n-1)}}$$

$$\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1} \ ⇒ \ S_5 \ = \ \frac{10(3^5 \ - \ 1)}{3 \ - \ 1} \ = \ 5(243 \ - \ 1) \ = \ 1210}$$

10) Find the explicit formula and the sum of the first $$10$$ terms: $$96, \ 48, \ 24, \ 12, \ 6, \ ...$$

$$\color{red}{a \ = \ 96, \ r \ = \ \frac{1}{2} \ , \ a_n = a \ r^{(n-1)}}$$
$$\color{red}{⇒ \ a_n = 96 \ (\frac{1}{2} \ )^{(n-1)}}$$

$$\color{red}{S_n \ = \ \frac{a(r^n \ - \ 1)}{r \ - \ 1}}$$

$$\color{red}{⇒ \ S_{10} \ = \ \frac{96((\frac{1}{2} \ )^{10} \ - \ 1)}{2 \ - \ 1} \ = \ \frac{96}{1023}}$$

## Geometric Sequences Practice Quiz

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