1. FM Series

2. BAT Understand and write series using sigma notation
Series: Proof
KUS objectives
BAT Understand and write series using sigma notation
BAT Generate sequences
BAT Solve arithmetic series problems involving sigma notation
1 𝑛 𝑟 = 𝑛 2 𝑛+1
1 𝑛 𝑟 2 = 𝑛 6 𝑛+1 (2𝑛+1)
1 𝑛 𝑟 3 = 𝑛 (𝑛+1) 2
Starter : Find
𝑟=1 32 𝑟 2
= (65)=34320
𝑟=11 44 𝑟
= 𝑟=1 44 𝑟 − 𝑟=1 10 𝑟 = − 10 2 (11)=891
𝑟= 𝑟 3
= 𝑟=1 30 𝑟 3 − 𝑟=1 12 𝑟 3 = − =210141

3. Show that n2 + n – 1056 = 0 and find the value of n
𝑟=1 𝑛 𝑟 = 528 W8
Given that
Show that n2 + n – 1056 = 0 and find the value of n
𝑟=1 𝑛 𝑟= 1 2 𝑛(𝑛+1)=528
... ⇒ 𝑛 2 +𝑛−1056=0
⇒ (𝑛−32)(𝑛+33)=0
⇒ 𝑛=32, 𝑛≠−33

4. Remember: and: sum of the first n natural numbers sum of the first
Notes: reminder
1 𝑛 4 =4𝑛
Remember:
1 𝑛 (𝑎𝑟+𝑏) =𝑎 1 𝑛 𝑟 +𝑏𝑛
and:
1 𝑛 𝑟
sum of the first n
natural numbers
= 1 2 𝑛(𝑛+1)
1 𝑛 𝑟 2 = 1 6 𝑛(𝑛+1)(2𝑛+1)
sum of the first
n squares is
1 𝑛 𝑟 3 = 1 4 𝑛 2 (𝑛+1 ) 2
sum of the first
n cubes is

5. Similarly: Notes (cont) 1 𝑛 (2 𝑟 3 +3 𝑟 2 +4𝑟+5)=
1 𝑛 (2 𝑟 𝑟 2 +4𝑟+5)=
2 1 𝑛 𝑟 𝑛 𝑟 𝑛 𝑟 +5𝑛

6. 1 𝑛 7𝑟−4 =7 1 𝑛 𝑟 −4(𝑛) =7 𝑛 2 (𝑛−1) −4(𝑛) = 𝑛 2 7𝑛+7−8 = 𝑛 2 7𝑛−1
WB9 algebra We might also derive a formula a) show that 1 𝑛 7𝑟−4 = 𝑛 2 (7𝑛−1)
1 𝑛 7𝑟−4 =7 1 𝑛 𝑟 −4(𝑛)
=7 𝑛 2 (𝑛−1) −4(𝑛)
= 𝑛 2 7𝑛+7−8
= 𝑛 2 7𝑛−1

7. 1 𝑛 𝑟 2 +2𝑟−1 = 1 𝑛 𝑟 2 +2 1 𝑛 𝑟 −𝑛 = 1 6 𝑛(𝑛+1)(2𝑛+1)+2 1 2 𝑛(𝑛+1) −𝑛
WBx algebra We might also derive a formula b) Show that: 1 𝑛 ( 𝑟 2 +2𝑟−1) = 1 6 𝑛(2 𝑛 2 +9𝑛+1)
1 𝑛 𝑟 = 𝑛 2 𝑛+1
1 𝑛 𝑟 2 = 𝑛 6 𝑛+1 (2𝑛+1)
1 𝑛 𝑟 3 = 𝑛 (𝑛+1) 2
1 𝑛 𝑟 2 +2𝑟−1 = 1 𝑛 𝑟 𝑛 𝑟 −𝑛
= 1 6 𝑛(𝑛+1)(2𝑛+1) 𝑛(𝑛+1) −𝑛
Here we look for common factors and do some ‘fixing’
= 1 6 𝑛[(𝑛+1)(2𝑛+1)+6(𝑛+1)−6]
= 1 6 𝑛(2 𝑛 2 +9𝑛+1)

8. WB10a algebra a) show that 1 2𝑛 𝑟 2 = 𝑛 6 (2𝑛+1)(7𝑛+1)
1 𝑛 𝑟 = 𝑛 2 𝑛+1
1 𝑛 𝑟 2 = 𝑛 6 𝑛+1 (2𝑛+1)
1 𝑛 𝑟 3 = 𝑛 (𝑛+1) 2
𝑛+1 2𝑛 𝑟 2 = 1 2𝑛 𝑟 2 − 1 𝑛 𝑟 2
= 2𝑛 6 (2𝑛+1)(4𝑛+1) − 𝑛 6 (𝑛+1)(2𝑛+1)
= 𝑛 6 (2𝑛+1) 2 4𝑛+1 −(𝑛+1)
= 𝑛 6 (2𝑛+1)(7𝑛+1)

9. b) Prove that WB10b 𝑟=5 2𝐾−1 𝑟 =2 𝐾 2 −𝐾−10, 𝐾≥3
𝑟=5 2𝐾−1 𝑟 =2 𝐾 2 −𝐾−10, 𝐾≥3
b) Prove that
1 𝑛 𝑟 = 𝑛 2 𝑛+1
1 𝑛 𝑟 2 = 𝑛 6 𝑛+1 (2𝑛+1)
1 𝑛 𝑟 3 = 𝑛 (𝑛+1) 2
𝑟=5 2𝐾−1 𝑟 = 𝑟=1 2𝐾−1 𝑟 − 𝑟=1 4 𝑟
= (2𝐾−1)(2𝐾) 2 − (4)(5) 2
...=2 𝐾 2 −𝐾−10

10. 𝑛+1 𝑛 𝑟 2 +𝑟−2 = 1 𝑛 𝑟 2 − 1 𝑛 𝑟 −2n = 𝑛 6 (𝑛+1)(2𝑛+1) − 𝑛 2 𝑛+1 −2𝑛
WB a) show that 1 𝑛 𝑟 2 +𝑟−2 = 𝑛 3 (𝑛+4)(𝑛−1) b) hence, find the sum of the series …+418
1 𝑛 𝑟 = 𝑛 2 𝑛+1
1 𝑛 𝑟 2 = 𝑛 6 𝑛+1 (2𝑛+1)
1 𝑛 𝑟 3 = 𝑛 (𝑛+1) 2
𝑛+1 𝑛 𝑟 2 +𝑟−2 = 1 𝑛 𝑟 2 − 1 𝑛 𝑟 −2n
= 𝑛 6 (𝑛+1)(2𝑛+1) − 𝑛 2 𝑛+1 −2𝑛
= 𝑛 6 𝑛+1 2𝑛+1 −3 𝑛+1 −12
= 𝑛 𝑛 2 +6𝑛−8
= 𝑛 3 ( 𝑛 2 +3𝑛−4)
= 𝑛 3 𝑛+4 𝑛−1 QED

11. WB 11b a) show that 1 𝑛 𝑟 2 +𝑟−2 = 𝑛 3 (𝑛+4)(𝑛−1) b) hence, find the sum of the series …+418
…+418= 𝑟 2 +𝑟−2
= (24)(19)
=3040

12. 𝑎) 𝑘=1 2𝑛−1 𝑘 = (2𝑛−1) 2 (2𝑛−1+1)= n(2n−1)
1 𝑛 𝑟 = 𝑛 2 𝑛+1
1 𝑛 𝑟 2 = 𝑛 6 𝑛+1 (2𝑛+1)
1 𝑛 𝑟 3 = 𝑛 (𝑛+1) 2
WB a) Find 𝑘=1 2𝑛−1 𝑘 b) Hence show that 𝑘=𝑛+1 2𝑛−1 𝑘 = 3𝑛 2 𝑛−1 , 𝑛≥2
𝑎) 𝑘=1 2𝑛−1 𝑘 = (2𝑛−1) 2 (2𝑛−1+1)= n(2n−1)
𝑏) 𝑛+1 2𝑛−1 𝑘 = 1 2𝑛−1 𝑘 − 1 𝑛 𝑘
=𝑛 2𝑛−1 − 𝑛 2 𝑛+1
= 3𝑛 𝑛− 2 3 − 3𝑛 2 𝑛
= 3𝑛 𝑛− 2 3 − 𝑛 3 − 1 3
= 3𝑛 2 𝑛− QED

13. 𝑎) 𝑟=1 𝑛 2 𝑟 − 𝑟=1 𝑛 𝑟 = 𝑛 2 2 𝑛 2 +1 − 𝑛 2 (𝑛+1)
1 𝑛 𝑟 = 𝑛 2 𝑛+1
1 𝑛 𝑟 2 = 𝑛 6 𝑛+1 (2𝑛+1)
1 𝑛 𝑟 3 = 𝑛 (𝑛+1) 2
WB a) show that 𝑟=1 𝑛 2 𝑟 − 𝑟=1 𝑛 𝑟 = 𝑛( 𝑛 3 −1) b) Hence evaluate 𝑟= 𝑟
𝑎) 𝑟=1 𝑛 2 𝑟 − 𝑟=1 𝑛 𝑟 = 𝑛 𝑛 2 +1 − 𝑛 2 (𝑛+1)
= 𝑛 2 𝑛 3 +𝑛 −𝑛 −1
= 𝑛( 𝑛 3 −1) 2
𝑏) 𝑟= 𝑟 = 𝑟 − 1 9 𝑟
= 9( 9 3 −1) 2 − 3( 3 3 −1) 2
=3237

14. WB 14 Given that 𝑢 𝑟 =𝑎𝑟+𝑏 and 𝑟=1 𝑛 𝑢 𝑟 = 𝑛 2 7𝑛+1
𝑟=1 𝑛 𝑎𝑟+𝑏 =𝑎 𝑟=1 𝑛 𝑟 +𝑏𝑛
=𝑎× 𝑛 2 𝑛+1 +𝑏𝑛
= 𝑛 2 𝑎𝑛+2𝑏𝑛+𝑎
= 𝑛 2 7𝑛+1
Equating gives 𝑎+2𝑏=7 and a=1
a=1, 𝑏=3

15. Crucial points
1. Check your results When you find a sum of the first n terms of a series, it is a good idea to substitute n = 1, and perhaps n = 2 as well, to check your result.
2. Look for common factors When using standard results, there can be quite a lot of algebra involved in simplifying the result. Make sure you take out any common factors first, as this makes the algebra a lot simpler.
3. Be careful when summing a constant term
remember
1 𝑛 𝑘 =𝑘+𝑘+𝑘 𝑘=𝑘𝑛

16. KUS objectives
BAT Understand and write series using sigma notation
BAT Generate sequences
BAT Solve arithmetic series problems involving sigma notation
self-assess
One thing learned is –
One thing to improve is –

17. END