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

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### 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 sequence has the following terms: a_{3}= 32, a

_{4}= 43, a

_{5}= 54, a

_{6}= 65, ...

What type of sequence is it? Find the first term and the common difference/ratio, and hence the 25

^{th}term (a

_{25}).

[Arithmetic, a

_{1}= -1, d = 11, a

_{25}= 263]

Ex. 1:

A geometric sequence has the following terms: u_{5}= 4.000, u

_{6}= -4.000, u

_{7}= 4.000, u

_{8}= -4.000, u

_{9}= 4.000, u

_{10}= -4.000, ...

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

_{inf}) of the series.

[u

_{1}= 4.00, r = -1.00, S

_{inf}= 2.00]

Ex. 2:

A sequence has the following terms: a_{6}= 6, a

_{7}= 4, a

_{8}= 2, a

_{9}= 0, ...

What type of sequence is it? Find the first term and the common difference/ratio, and hence the 37

^{th}term (a

_{37}).

[Arithmetic, a

_{1}= 18, d = -2, a

_{37}= -54]

Ex. 3:

A sequence has the following terms: u_{5}= -2.09, u

_{6}= -1.32, u

_{7}= -0.84, u

_{8}= -0.53, u

_{9}= -0.34, u

_{10}= -0.21, ...

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}= -13.00, r = 0.63, u

_{16}= -0.01]

Ex. 4:

A geometric sequence has the following terms: u_{3}= 0.889, u

_{4}= 0.296, u

_{5}= 0.099, u

_{6}= 0.033, u

_{7}= 0.011, u

_{8}= 0.004, ...

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

_{inf}) of the series.

[u

_{1}= 8.00, r = 0.33, S

_{inf}= 12.00]

Ex. 5:

A sequence has the following terms: u_{4}= 4.63, u

_{5}= -3.86, u

_{6}= 3.22, u

_{7}= -2.68, u

_{8}= 2.23, u

_{9}= -1.86, ...

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}= -8.00, r = -0.83, u

_{15}= -0.62]

Ex. 6:

A sequence has the following terms: u_{3}= 0.17, u

_{4}= -0.03, 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 12

^{th}term (a

_{12}).

[Geometric, u

_{1}= 6.00, r = -0.17, u

_{12}= -0.00]

Ex. 7:

A geometric sequence has the following terms: u_{3}= -0.680, u

_{4}= 0.136, u

_{5}= -0.027, u

_{6}= 0.005, u

_{7}= -0.001, u

_{8}= 0.000, ...

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

_{inf}) of the series.

[u

_{1}= -17.00, r = -0.20, S

_{inf}= -14.17]

Ex. 8:

A sequence has the following terms: u_{3}= -0.32, u

_{4}= -0.06, u

_{5}= -0.01, 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 12

^{th}term (a

_{12}).

[Geometric, u

_{1}= -8.00, r = 0.20, u

_{12}= -0.00]

Ex. 9:

A geometric sequence has the following terms: u_{3}= -1.188, u

_{4}= 0.297, u

_{5}= -0.074, u

_{6}= 0.019, u

_{7}= -0.005, u

_{8}= 0.001, ...

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

_{inf}) of the series.

[u

_{1}= -19.00, r = -0.25, S

_{inf}= -15.20]