12 | February | 2018 | Alex's Very Professional Blog

This week, I focused on geometric sequences.

Geometric sequences share similar characteristics to arithmetic sequences. For one, geometric sequences are patterns, just like our arithmetic sequences from last week! Arithmetic sequences have common differences, such as this one: “5, 10, 15, 20, 25, 30…” You can tell the pattern here is that 5 is being added for every new term!

Geometric sequences, however, have common ratios. Like so: “2, 4, 8, 16, 32, 64…” You should be able to see a pattern here. What’s the pattern? The terms are being multiplied by 2 each time! This is very different from our arithmetic sequence, where numbers were being added as the pattern was continuing. The ratio is determined like so: tn/(tn-1)=r.

The ratio is a very important number. It’s as important to geometric sequences as the common difference, d, is to arithmetic sequences, so get used to finding it!

There are a number of ways that you can utilize the common ratio, r. Say, you wanna find the 9th term in our geometric sequence here. Looks something like this:

t1*r^(n-1) = t9

2*2^(9-1) = t9

2*2^8 = t9

2*256 = t9

512 = t9

The general formula for geometric sequences is t1*r(n-1)=tn

First week of Pre-Calc 11! This week, I learned about Arithmetic Sequences.

Arithmetic sequences are like patterns, they look something like this: 5, 7, 9, 11, 13….

For a pattern to be an arithmetic sequence, all the numbers must have a common difference between them all. The common difference is how much it goes up or down. For our pattern here, the common difference is 2! You see how the pattern continues to increase by 2? This is an arithmetic sequence.

If you wanted to find the 20th term in that pattern, you would use the following formula:

$t_10=t_1+9d$

$t_10 = 5 + 9(2)$

$t_10 = 5 + 18$

$t_10 = 23$

With this formula, we’re saying that we’re going to take the first term in the arithmetic sequence, add the common difference (2) NINE times to get the 10th term. Let’s count it, and make sure my work is correct.

5, 7, 9, 11, 13, 15, 17, 19, 21, 23

Look at that, the 10th term really is 23! For this formula, you don’t have to use t1. You can use any term you already have. If we only had t4, the formula would look like this: