Sequences + Series

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What are sequences and series?

A sequence is a list of numbers, such as: 1, 2, 3, 4, 5, 6. Each number in the sequence is a term.

Later, we can name these terms such as  

for the first, second, third, fourth, fifth term.

By adding the terms in a sequence, this results in a series:

1 + 2 + 3 + 4 + 5 + 6

Arithmetic Sequences:

There is a common difference between each of the terms:

  • 1, 2, 3, 4, 5, 6: the common difference between each term is 1.
  • 2, 5, 8, 11, 14: the common difference between each term is 3.

General Term:

The general term is a formula for finding any term in a sequence. This continually addition is a pattern -we can turn that into an equation to find the nth term in a sequence, whatever n may be.

General Term of an Arithmetic Sequence:

  • d = common difference between terms
  • n = index of term you want to find – if you want to find 3rd term, then n is 3

We can start by writing out an arithmetic sequence and then converting it to an sequence with only variables:

1, 3, 5, 7, 9

We note that the first term is 1, the common difference d = 2, and let’s say that we want to find the 6th term, so n = 6.

If we were to convert the above sequence to only have variables:

If we look at how to get from 1 to 11, we note that we need to add 2 five times, where 11 is the 6th term of the sequence. We can notice to get the nth term (6th) of a sequence, we need to times the common difference by n – 1 (2 times 5) and add the first term (1):

forming the general term of an arithmetic sequence.

Calculating the sum of an arithmetic series:

As we mentioned before, a series is the addition of a sequence:

1 + 2 + 4 + 8 + 16

Therefore, an arithmetic series is the addition of an arithmetic sequence:

1 + 3 + 5 + 7 + 9

To find the sum of an arithmetic sequence, once again, we can create a formula to compute these sums regardless of which numbers are the first and last terms. A famous problem for calculating the sum of an arithmetic series is finding the first 100 integers:

1 + 2 + 3 + 4 + 5 + … + 99 + 100 = ?

You could simply brute force through this problem; add each number individually, but there is a much simpler and efficient method. By adding the first and last term, then the 2nd term and the 2nd last term, and so on, you get:

1 + 100 = 101

2 + 99 = 101

3 + 98 = 101

Clearly, each pair added together equals 101 –so to find the sum– how many pairs are there in total?

There are 100 numbers, and therefore 100/2 = 50 pairs. So 50 * 101 = 5050 – the sum of the first 100 integers.

I hope you enjoyed this week’s post, thank you for reading, and I hope you have a great day!

Sources:

  • Introduction to Algebra: Art of Problem Solving