1. nth or General Term of an Arithmetic Sequence
Sequences and Series
nth or General Term of an Arithmetic Sequence
where a1 is the first term of the sequence and d is the common difference.
Where d = a2-a1 = a3-a2

2. nth or General Term of a Geometric Sequence
tn = arn-1
where a is the first term of the sequence and r is the common ratio.
Where r = t2÷ t1 = t3÷ t2

3. If, as n increases, tn approaches a value ℓ more and more closely
and the sequence has a limit ℓ
If, as n increases, | tn | increases without limit, or tn oscillates,
and the sequence has no limit.

4. For the arithmetic series a + (a + d) + (a + 2d) +…
the sum of n terms is given by
Sn = [ 2a + (n – 1) d ] Or
Sn = (t1 + tn)

5. For the geometric series a + ar2 + ar3 +…
the sum of n terms is given by or
Use this when
r > 1
Use this when
r < 1

6. Infinite series – the limit of a sum
A series may have a limit
For some series, as more terms are added on, the sums produced approach nearer and nearer to a particular value. The value ℓ is called the SUM TO INFINITY of the series and we say the series is CONVERGENT.

7. Infinite series – the limit of a sum
If, as more terms are added Sn gets larger or more negative without approaching a particular value, or oscillates, then the series does not have a limit and we say it is DIVERGENT.

8. Any arithmetic series is divergent.
Limit of a geometric series:
Not all geometric series have a limit
If r ≥ 1, Sn increases without end
If r ≤ -1, Sn oscillates.
If -1 < r < 1 Sn will have a limit and the series is convergent

9. Limit of a geometric series formula: