## Physics and Maths (STEM) Tutor - coaching for all - from GCSE through A-Level to Degree

## Other tutor / coaching services: GCSE/A-level Computer Science

### Proper, Improper and Mixed Fractions

### Fractional Arithmetic

### Changing the Subject of an Expression

### Multiplying Expressions

### Factorising Quadratic Expressions

### Solving Linear Equations

### Rationalising the Denominator

### Solving Simultaneous (Linear) Equations

### Solving Simultaneous Quadratic Equations

### 2 and 3 Part Ratios

### Completing the Square

### Sequences and Series

Ex. 0:

A geometric sequence has the following terms: u_{5}= 0.001, u

_{6}= -0.000, u

_{7}= 0.000, u

_{8}= -0.000, u

_{9}= 0.000, u

_{10}= -0.000, ...

Find the first term and the common ratio, and hence the sum to infinity (S

_{inf}) of the series.

[u

_{1}= 5.00, r = -0.13, S

_{inf}= 4.44]

Ex. 1:

A sequence has the following terms: a_{7}= -125, a

_{8}= -141, a

_{9}= -157, a

_{10}= -173, ...

What type of sequence is it? Find the first term and the common difference/ratio, and hence the 29

^{th}term (a

_{29}).

[Arithmetic, a

_{1}= -13, d = -16, a

_{29}= -461]

Ex. 2:

A sequence has the following terms: a_{7}= -83, a

_{8}= -96, a

_{9}= -109, a

_{10}= -122, ...

What type of sequence is it? Find the first term and the common difference/ratio, and hence the 56

^{th}term (a

_{56}).

[Arithmetic, a

_{1}= 8, d = -13, a

_{56}= -707]

Ex. 3:

A sequence has the following terms: u_{3}= 0.02, u

_{4}= -0.00, u

_{5}= 0.00, u

_{6}= -0.00, u

_{7}= 0.00, u

_{8}= -0.00, ...

What type of sequence is it? Find the first term and the common difference/ratio, and hence the 15

^{th}term (a

_{15}).

[Geometric, u

_{1}= 14.00, r = -0.03, u

_{15}= 0.00]

Ex. 4:

A sequence has the following terms: u_{4}= -1.46, u

_{5}= -0.63, u

_{6}= -0.28, u

_{7}= -0.12, u

_{8}= -0.05, u

_{9}= -0.02, ...

What type of sequence is it? Find the first term and the common difference/ratio, and hence the 13

^{th}term (a

_{13}).

[Geometric, u

_{1}= -18.00, r = 0.43, u

_{13}= -0.00]

Ex. 5:

A sequence has the following terms: u_{3}= 1.76, u

_{4}= 0.70, u

_{5}= 0.28, u

_{6}= 0.11, u

_{7}= 0.05, u

_{8}= 0.02, ...

What type of sequence is it? Find the first term and the common difference/ratio, and hence the 15

^{th}term (a

_{15}).

[Geometric, u

_{1}= 11.00, r = 0.40, u

_{15}= 0.00]

Ex. 6:

A geometric sequence has the following terms: u_{4}= 0.088, u

_{5}= 0.018, u

_{6}= 0.004, u

_{7}= 0.001, u

_{8}= 0.000, u

_{9}= 0.000, ...

Find the first term and the common ratio, and hence the sum to infinity (S

_{inf}) of the series.

[u

_{1}= 11.00, r = 0.20, S

_{inf}= 13.75]

Ex. 7:

A sequence has the following terms: u_{4}= 3.09, u

_{5}= 1.75, u

_{6}= 0.99, u

_{7}= 0.56, u

_{8}= 0.32, u

_{9}= 0.18, ...

What type of sequence is it? Find the first term and the common difference/ratio, and hence the 15

^{th}term (a

_{15}).

[Geometric, u

_{1}= 17.00, r = 0.57, u

_{15}= 0.01]

Ex. 8:

A sequence has the following terms: u_{5}= -0.56, u

_{6}= -0.28, u

_{7}= -0.14, u

_{8}= -0.07, u

_{9}= -0.04, u

_{10}= -0.02, ...

What type of sequence is it? Find the first term and the common difference/ratio, and hence the 16

^{th}term (a

_{16}).

[Geometric, u

_{1}= -9.00, r = 0.50, u

_{16}= -0.00]

Ex. 9:

A sequence has the following terms: u_{4}= 1.75, u

_{5}= 0.88, u

_{6}= 0.44, u

_{7}= 0.22, u

_{8}= 0.11, u

_{9}= 0.05, ...

What type of sequence is it? Find the first term and the common difference/ratio, and hence the 16

^{th}term (a

_{16}).

[Geometric, u

_{1}= 14.00, r = 0.50, u

_{16}= 0.00]