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
