1. Geometric Series

2. Write the expression for the nth term of an arithmetic series
Geometric Series
KUS objectives
BAT work out the sum of a Geometric Sequence
BAT solve problems using the formula for Sn
Starter:
Write the expression for the nth term of an arithmetic series
Write the formula for the sum of an arithmetic series
Write the instructions for the proof of the formula for the sum of an arithmetic series

3. Follow these instructions:
Wb9 Proof
Follow these instructions:
Write Sn = the first five and last algebraic terms of a geometric series
Write the result of the first line with every term multiplied by the common ratio (r)
Subtract the 2nd line from the first
Factorise the LHS and RHS
Rearrange so that LHS is Sn 1
Multiply all terms by r 2
Factorise both sides
Divide by (1 - r)
𝑆 𝑛 = 𝑎 𝑟 𝑛 −1 𝑟−1

4. WB10 Using the formula
𝑺𝒏= 𝒂(𝟏−𝒓𝒏) 𝟏−𝒓
𝑻𝒏=𝒂 𝒓 𝒏−𝟏
a) A geometric sequence has first term 20 and common ratio ¾
Find the sum of the first ten terms of the series,
Giving your answer to 3 dp
𝑆 10 = 20 1− − =75.495
𝑎=20 𝑟= 𝑛=10
b) A geometric sequence has first term 108 and common ratio 5 4
Find the sum of the first twelve terms of the series,
Giving your answer to 3 dp
𝑆 12 = − −1 =
𝑎= 𝑟= 𝑛=12

5. Show that the common ratio of the sequence is 3 5
WB11 Using the formula
𝑺𝒏= 𝒂(𝟏−𝒓𝒏) 𝟏−𝒓
𝑻𝒏=𝒂 𝒓 𝒏−𝟏
The third term of a geometric series is 5625 and the sixth term is 1215
Show that the common ratio of the sequence is 3 5
Find the first term of the sequence
Find the sum of the first 18 terms of the sequence

6. Find the sum of the following series:
WB 12a find n using logs
𝑺𝒏= 𝒂(𝟏−𝒓𝒏) 𝟏−𝒓
𝑻𝒏=𝒂 𝒓 𝒏−𝟏
Find the sum of the following series:
… + … + 1
𝒂=𝟏𝟎𝟐𝟒, 𝒓=𝟎.𝟓, 𝒏= ?
1= 1024(0.5) 𝑛−1
𝑆 𝑛 = 𝑎 1− 𝑟 𝑛 1−𝑟
= (0.5) 𝑛−1
‘Take logs’
Sub in all values
𝑙𝑜𝑔 =𝑙𝑜𝑔 𝑛−1
𝑆 𝑛 = − −0.5
Calculate
𝑙𝑜𝑔 = 𝑛−1 log⁡(0.5)
𝑆 𝑛 =2047
𝑛 = 11

7. Find the sum of the following series:
WB 12b Counter example
𝑺𝒏= 𝒂(𝟏−𝒓𝒏) 𝟏−𝒓
𝑻𝒏=𝒂 𝒓 𝒏−𝟏
Find the sum of the following series:
… + … + 1
𝒂=𝟏𝟎𝟐𝟒, 𝒓=− 𝟎.𝟓, 𝒏= ?
1= 1024(−0.5) 𝑛−1
= (−0.5) 𝑛−1
log⁡(−0.5)
You cannot take logs of a negative value…
𝑙𝑜𝑔 =𝑙𝑜𝑔 −0.5 𝑛−1
𝑙𝑜𝑔 = 𝑛−1 log⁡(−0.5)
𝐸𝑟𝑟𝑜𝑟‼
Why can’t we work this out?

8. Find the sum of the following series:
WB 12b Counter example continued
𝑺𝒏= 𝒂(𝟏−𝒓𝒏) 𝟏−𝒓
𝑻𝒏=𝒂 𝒓 𝒏−𝟏
Find the sum of the following series:
… + … + 1
𝒂=𝟏𝟎𝟐𝟒, 𝒓=− 𝟎.𝟓, 𝒏= ?
1= 1024(−0.5) 𝑛−1
1= 1024(0.5) 𝑛−1
If we work through as before, using logs,
we will get the same answer for n (11)
Remember at this point that you are just working out the number of terms…
The negative value here does not affect how many terms there are, it just makes the pattern alternate between positive and negative numbers
So if it were just ‘0.5’ rather that ‘-0.5’, we will still have the correct answer for n (the number of terms)
So you can actually just use 0.5 for now, as it is only the absolute value that is important here
(The negative will be important when we come to work out the sum of the sequence…)
𝑆 𝑛 = 𝑎 1− 𝑟 𝑛 1−𝑟
𝑆 𝑛 = − (−0.5) −(−0.5)
𝑆 𝑛 =683
 The negative value of -0.5 DOES affect the sum of the sequence, so we do need to include it now!

9. So the least number of terms is n = 21
WB 13 Sums and using logs for nth term
𝑺𝒏= 𝒂(𝟏−𝒓𝒏) 𝟏−𝒓
𝑻𝒏=𝒂 𝒓 𝒏−𝟏
a) Find the value of n at which the sum of the following
sequence is greater than 2,000,000
… + … +
𝑎=1 𝑟=2
𝑺𝒏= 𝟏( 𝟐 𝒏 −𝟏) 𝟐−𝟏 >
Becomes 𝟐 𝒏 >
Take logs 𝒏> 𝑙𝑜𝑔 =20.9
So the least number of terms is n = 21

10. So the least number of terms is n = 8
WB 13b Sums and using logs for nth term
𝑺𝒏= 𝒂(𝟏−𝒓𝒏) 𝟏−𝒓
𝑻𝒏=𝒂 𝒓 𝒏−𝟏
b) The sum of the geometric series … exceeds find the least number of terms
𝑎=2 𝑟=3
𝑺𝒏= 𝟐( 𝟑 𝒏 −𝟏) 𝟑−𝟏 > 5000
Becomes 𝟑 𝒏 > 5001
Take logs 𝒏> 𝑙𝑜𝑔 =7.753
Answers
3. R = ¾ a = 288
So the least number of terms is n = 8

11. So the least number of years is n = 19
WB14 percentages exam Q
Tom saves money every year. The first year he saves £100. Each year he increases the amount he put in the bank and saves by 10%. After how many years do Tom’s savings exceed £6000 (excluding any interest he has earned)
𝑎= 𝑟=1.1
𝑺𝒏= 𝟏𝟎𝟎( 𝟏.𝟏 𝒏 −𝟏) 𝟏.𝟏−𝟏 >6000
Becomes 𝟏.𝟏 𝒏 >6000÷1000=6
Take logs 𝒏> 𝑙𝑜𝑔 =
So the least number of years is n = 19

12. One thing to improve is –
KUS objectives
BAT work out the sum of a Geometric Sequence
BAT solve problems using the formula for Sn
self-assess
One thing learned is –
One thing to improve is –

13. END