Sequences revision
In a nutshell
General sequences are a list of numbers which can either be described by a term to term rule or position to term rule. These sequences have various patterns and may or may not have limits.
Defining a sequence
The term to term rule gives a term in a general sequence and the calculation to go to the next term. The position to term rule uses an equation to work out a number in a sequence if the position in the sequence is known.
Example 1
Given that u1=10 and un+1=2un+3, what is the 3rd term in the sequence?
The number n indicates the position in the sequence. So u1 is the first term in the sequence. To find u2 (the second term), substitute n=1 into the equation of the sequence:
u1+1u2=2u1+3=2(10)+3
u2=23
Substitute n=2 into the equation of the sequence to find the third term:
u2+1u3=2u2+3=2(23)+3
u3=49
Example 2
What is the 6th term in a sequence with the formula u2n=24n+5?
Find the value of n which will give the 6th term:
u2n2nn=u6=6=3
Substitute n=3 into the formula:
u2(3)=24(3)+5
u6=217=8.5
Increasing, decreasing or periodic sequences
Sequences may increase such that each term is always larger than the previous one, they may decrease such that each term is smaller than the previous one or they may be periodic such that the sequence eventually repeats itself.
Example 3
Find the first 4 terms of un+1=un1 given that u1=5. Describe the behaviour of this sequence.
Calculate u2, u3, u4:
u2u3u4=51=511=5=51
Describe the behaviour of the sequence:
The sequence is periodic with a periodicity of 2.
Note: Periodicity means how many terms there are that the sequence goes through before repeating.
Limits
A given general sequence may be convergent or divergent. A convergent sequence has a limit such that as n→∞ ("n tends to infinity") the values of un and un+1 will be the same. A divergent sequence has no limit, so as n→∞ the values of un and un+1 will continue to be different.
Example 4
A convergent sequence is defined as un+1=53un+4. What is the limit, L?
Consider n→∞. You know that the sequence is convergent so there exists a limit L such that
un+1=un=L
Using the sequence equation, solve for L:
L=53(L)+4
52(L)=4
2L=20L=10