What you need to know

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

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

4.  common ratio = √2 10th term = 64 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 • a = 4 √2 substituting into first equation gives a √2 = 4 • a = 4√2 2  First term = 2√2 10th term = ar 9 = 2 (2 ) 5 = 2 ( √ 2 ) 10 = 2 6 = 2√2 ( √ 2 ) 9

5. Example 2 How many terms are there in the geometric sequence 0.2, 1, 5, …………, 390625? 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 3 x a r 9 10th term = x 2 = 3 ( x + 6 ) = 3 x 2 9 x 2 = 3 x + 18  10th term is 1536 x 2 - 3 x - 18 = 0 ( x – 6 ) ( x + 3 ) = 0 x = 6 or -3 x = 6 ( since there are no negative terms )

7. S  = a 1 - r = 6 Infinite Geometric Series Consider the series S = 3 + 1.5 + 0.75 + 0.375 + 0.1875 + ….. Summing the terms one by one gives 3, 4.5, 5.25, 5.625, 5.8125, … 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. ( The limit only exists if -1 < r < 1 ) So for above example S  = 3 1 – 0.5

8. Example 4 The first term of a GP is 10 and the common ratio is 0.8. (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. Show the inequality S – SN < 0.01 can be written as 0.8N < 0.0002 and use logarithms to find the smallest possible value of N Smallest N is 39

10. Pascal’s triangle 1 1 1 2 1 1 3 3 1 1 4 6 4 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=1 + 8x + 24x2+ 32x3 +16x4 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. Example 7 Find the coefficient of t4 in the expansion of (3 – 2t)8 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

14. Example 9 Find the Binomial expansion of (2x + 5)4 , simplifying the terms

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.

18. 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