- 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.
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.
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)
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).
n = the number of terms
a1 = first term (a)
d = common difference
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
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
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
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:
|Proof for the Sum of Arithmetic Series|