2.3 Pure - Sequences

Arithmetic sequence

Where a is the first term, d is the common difference

a + (n-1)d

Sum of an arithmetic sequence

n/2 (2a + (n-1)d)

Proof of the sum of an arithmetic sequence

1) S_n = a + (a+d) + (a+2d) + …

2) S_n = (a + (n-1)d) + (a + (n-2)d) + ….

1 + 2

2S_n = n(2a + (n-1)d)

Geometric sequence

ar^n-1

Where r is the common ratio

Sum of a geometric sequence

(a(rⁿ-1)) / (r-1)

Proof of sum of geometric sequence

1) S_n = a + ar + ar² + ar³ + … ar^n-2 + ar^n-1

2) rS_n = ar + ar² + ar³ + … + ar^n-1 + arⁿ

1 - 2

S_n - rS_n = a - arⁿ

Sum to infinity for geometric sequences

S_infinity = a / (1 - r)

Only to be used for convergent sequences were /r/ < 1

Sigma notation

n SIGMA r = x

x = start value

n = number of times

Example of sigma notation