Sequences and Series

•  Edexcel - C1, Sequences and Series
• AQA - C2, Sequences and Series
• OCR - C2, Sequenes and Series

A sequence is a list of numbers, in it's basic definition. This list can be of a finite / infinite length.e.g. 2,4,6,8.. is a sequence of all positive even numbers. There are many types of sequences, we shall be looking at the Arithmetic Series / Progression.

A series is the sum of the terms in a sequence. Again these can be finite and infinite, depending on the sequence itself.

Arithmetic Sequences

An Arithmetic Sequence is a sequence of numbers, such that difference between the terms is a constant. e.g. 5,9,13,17,21 .. this difference here is + 4. This difference is called the common difference.


Each number in the sequence, is called a term. We call the first term (u1), second term (u2), third term (u3)... and so on. This is just notation. In the previous sequence example, 5 would be the first term, 9 the second, 13 the third.. and so on. The first term of a sequence is a. While the common difference is d.

Nth term, is a rule for finding any term in the sequence. Say if i wanted to find the 28th term of the previous sequence, it would be very long to add 4 each time, to get till the 28th term. Instead we can form a rule to find any term in that sequence. The formula for finding the nth term of a sequence is :

U(n) = a + (n-1)d

a = first term
d= common difference
n = the term you're finding

*This formula will work for any Arithmetic Sequence.

How we got this formula ?
As you know the first term is a. If each term goes up by a common difference, the second term must be a + d. The third term must be a + d + d = a + 2d, The fourth term must be a + d + d + d = a +3d.... and so on... if we look for the nth term it must be a + (n-1)d.


Example
a) Say we have an arithmetic sequence with the first term being 9. The common difference is -4. Find the 80th term ?

Here a is 9, d = -4 , and n=80

Use the formula U(80) = 9 + (80-1)*-4
=9 + (79*-4)
= 9+-(316)
= -307

Arithmetic Series

We also need to be able to find the Sum of an Arithmetic Sequence. There is a formula, we also need to be able to prove that formula (abit confusing, i'll include it in my image notes).

S(n) = The sum of n terms
n = the number of terms
a1 = first term (a)
d = common difference

Example

2,5,8,11,14,17....
a) Find the sum of the first ten terms
b) Find the sum of the terms starting from the 11th term and ending with the 28th term.

Identify a and d.
a = 2, d = +3

a) Use the Formula : 
S(10) =  10/2 [ 2(2) + (10-1)3]
= 5 [4 + 9(3)]
= 5 [4 + 27]
= 5 *31
=155

b) Use the Formula, note a is different.

They want to start from the 11th term, so the 11th term will be a.
 Use the nth term formula to find the 11th term : (Here we use a as 2)

=2 + (11-1)*3
= 2 + (10)*3
= 2 + 30
=32

11th term is 32, which is the a

Count how many terms are from 11 to 28. (18 terms), Now use the formula :
S(18) = 18/2 [ 2(32) + (18-1)*3]
= 9 [64 + 17*3]
= 9 [64 + 51]
= 9 x115
= 1035

Sigma Notation

The sigma sign Σ, is another notation you need to be able to interpret. It is simply the summation of an arithmetic sequence. I will use an example to make you understand it:

Here is an example, for sigma notation. The actual sign Σ means the sum. Now k^2, is the function / rule. The first term we have to add is below the sign, which is 2. So a =2, we have to add all the terms from 2 till 6. The number above the sign, is where we end. So the sum is 2 squared + 3 squared + 4 squared + 5 squared + 6 squared.. = 90.

Proof for the Sum of Arithmetic Series