1. What you need to know
To recognise GP’s and use nth term and sum of n terms formulae to solve problems
To know about the sum of an infinite GP where
How to use the binomial expansion of

2. Geometric Sequences eg. 10 20 40 80 81 27 9 3 x 2 x 2 x 2 x ⅓ x ⅓ x ⅓
x x x 2
x ⅓ x ⅓ x ⅓
first term = 10
first term = 81
common ratio = 2
common ratio = ⅓

3. Geometric kth term = a r k - 1 Term S n = a ( 1 – r n ) ( 1 – r )
Sum of terms
S n = a ( 1 – r n )
( 1 – r )
S  = a
1 - r
Sum to infinity

4. Find the common ratio, the first term and the 10th term
Example 1 Given that the 2nd term of a positive geometric sequence is 4 and the 4th term is 8.
Find the common ratio, the first term and the 10th term
2nd term = 4 
ar = 4
Ignore the negative root
4th term = 8 
ar 3 = 8
dividing gives r 2 = 2
 common ratio = √2
substituting into first equation gives a √2 = 4
a = 4 √2
a = 4√2 2
 First term = 2√2
10th term = ar 9
= 2√2 ( √ 2 ) 9
= 2 ( √ 2 ) 10
= 2 (2 ) 5
= 2 6
10th term = 64

5. Example 2 How many terms are there in the geometric sequence 0
nth term = a r n - 1
 There are 10 terms

6. Example 3 The numbers 3, x and ( x + 6 ) form the first three terms of a positive geometric sequence. Find the possible values of x and the 10th term of the sequence.
x = x + 6 x
10th term =
a r 9
x 2 = 3 ( x + 6 )
= 3 x 2 9
x 2 = 3 x + 18
 10th term is 1536
x x = 0
( x – 6 ) ( x + 3 ) = 0
x = 6 or -3
x = 6
( since there are no negative terms )

7. Infinite Geometric Series
Consider the series S = …..
Summing the terms one by one gives
3, 4.5, 5.25, 5.625, , …
No matter how many terms you take, the sum never exceeds a certain number.
We call this number the limit of the sum. We say the series is convergent.
S  = a
1 - r
( The limit only exists if -1 < r < 1 )
So for above example S  = 3
1 – 0.5
= 6

8. Example 4 The first term of a GP is 10 and the common ratio is 0. 8
Example 4 The first term of a GP is 10 and the common ratio is (a) Find the 4th term and the sum of the first 20 terms

9. (b) The sum of the first N terms is SN and the sum to infinity is S
(b) The sum of the first N terms is SN and the sum to infinity is S. Show the inequality S – SN < 0.01 can be written as 0.8N < and use logarithms to find the smallest possible value of N
Smallest N is 39

10. This button is on your calculator
Pascal’s triangle
1 1
1 2 1
Binomial expansion
(a+b)n can be expanded by using the numbers from Pascal’s triangle,
decreasing powers of a and increasing powers of b
This button is on your calculator

11. Example 6 Expand (1+2x)4 in ascending powers of x
(1+2x)4 = x x x3 +16x4 1 4 6 14 13 12 11 10
(2x)0
(2x)1
(2x)2
(2x)3
(2x)4
Put the powers of the first term descending
Put the numbers from Pascal’s triangle in first
Put the powers of the other term ascending

12. Put the numbers from Pascal’s are on the calculator
Example 7 Find the coefficient of t4 in the expansion of (3 – 2t)8 1 8 38 37 34
(-2t)0
(-2t)1
(-2t)4
Put the numbers from Pascal’s are on the calculator
t4 term is …90720t4 …..Coefficient of t4 is 90720

13. Example 8 Expand (1-x)5 as far as the term in x2.
Hence find an approximation to 0.95 1 5 10 15 14 13
(-x)0
(-x)1
(-x)2

14. Example 9 Find the Binomial expansion of (2x + 5)4 , simplifying the terms1 4 6
(2x)4
(2x)3
(2x)2
(2x)1
(2x)0 50 51 52 53 54

15. Example 9 Find the Binomial expansion of (2x + 5)4 , simplifying the terms
Hence show that (2x + 5)4 – (2x – 5)4 can be written as 320x3 + kx where the value of the constant k is to be stated.
k =2000

16. Verify that x = 2 is a root of the equation (2x + 5)4 – (2x – 5)4 = 3680x – 800 and find the other possible values of x.

17. x
4x3 -2

18. 4x2 8x -5 x
4x3
8x2
-5x -2
-8x2
-16x 10
Now factorise the quadratic factor

20. Summary
To recognise GP’s and use nth term and sum of n terms formulae to solve problems
To know about the sum of an infinite GP where
How to use the binomial expansion of