Patterns Revision Summary
Sequences and Series
Arithmetic Sequence or Linear Pattern with a constant 1st difference
n
d T2 T1 T3 T2 Tn a (n 1)d Sn (2a (n 1)d )
2
Geometric Sequence or Exponential Pattern with a constant ratio
T2 T3 a (1 r n ) a
r n 1 Sn S if 1 r 1
T1 T2 Tn a.r 1 r 1 r
Sigma Notation: Used for the sum of a series
Recognize the different forms for an AP and a GP
20 20
k 1
2k 4 4.2
k 1 is an AP k 1 is a GP
2
Quadratic Pattern: Tn an bn c
2a= 2nd Difference 3a +b = The first 1st difference a + b + c = 1st term
Always write out the pattern and show at least four 1st differences and three 2nd
differences
Combined Patterns:
Every alternate term (odd and even terms) form 2 different series
The sum to 100 terms would be the sum to 50 terms of each series
Types of Questions:
1) Given the first 3 terms enables you to find a and either d or r and any Tn or Sn
2) Given 2 terms enables you to find the sequence by solving simultaneous equations
3) Given a term or the sum and either a or d (or r) enables you to find n, the term
number or number of terms in the sum.
4) To find for which values of x a geometric series converges, make 1 r 1
where r is written in terms of x and solve for x
Remember : GP’s use exponents and logs and AP’s use linear and quadratic questions
Sequences and Series
Arithmetic Sequence or Linear Pattern with a constant 1st difference
n
d T2 T1 T3 T2 Tn a (n 1)d Sn (2a (n 1)d )
2
Geometric Sequence or Exponential Pattern with a constant ratio
T2 T3 a (1 r n ) a
r n 1 Sn S if 1 r 1
T1 T2 Tn a.r 1 r 1 r
Sigma Notation: Used for the sum of a series
Recognize the different forms for an AP and a GP
20 20
k 1
2k 4 4.2
k 1 is an AP k 1 is a GP
2
Quadratic Pattern: Tn an bn c
2a= 2nd Difference 3a +b = The first 1st difference a + b + c = 1st term
Always write out the pattern and show at least four 1st differences and three 2nd
differences
Combined Patterns:
Every alternate term (odd and even terms) form 2 different series
The sum to 100 terms would be the sum to 50 terms of each series
Types of Questions:
1) Given the first 3 terms enables you to find a and either d or r and any Tn or Sn
2) Given 2 terms enables you to find the sequence by solving simultaneous equations
3) Given a term or the sum and either a or d (or r) enables you to find n, the term
number or number of terms in the sum.
4) To find for which values of x a geometric series converges, make 1 r 1
where r is written in terms of x and solve for x
Remember : GP’s use exponents and logs and AP’s use linear and quadratic questions
