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

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

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.

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)

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

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.

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.

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

Limit of a geometric series formula: