FM Series

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 2 4 (𝑛+1) 2 Starter : Find 𝑟=1 32 𝑟 2 = 32 6 33 (65)=34320 𝑟=11 44 𝑟 = 𝑟=1 44 𝑟 − 𝑟=1 10 𝑟 = 44 2 43 − 10 2 (11)=891 𝑟=13 30 𝑟 3 = 𝑟=1 30 𝑟 3 − 𝑟=1 12 𝑟 3 = 30 2 4 31 2 − 12 2 4 1 3 2 =210141

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

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

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

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

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 2 4 (𝑛+1) 2 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 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)

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 2 4 (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)

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 2 4 (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

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 11 a) show that 1 𝑛 𝑟 2 +𝑟−2 = 𝑛 3 (𝑛+4)(𝑛−1) b) hence, find the sum of the series 4+10+18+28+40+…+418 1 𝑛 𝑟 = 𝑛 2 𝑛+1 1 𝑛 𝑟 2 = 𝑛 6 𝑛+1 (2𝑛+1) 1 𝑛 𝑟 3 = 𝑛 2 4 (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 6 2 𝑛 2 +6𝑛−8 = 𝑛 3 ( 𝑛 2 +3𝑛−4) = 𝑛 3 𝑛+4 𝑛−1 QED

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 4+10+18+28+40+…+418 0+4+10+18+28+40+…+418= 1 20 𝑟 2 +𝑟−2 = 20 3 (24)(19) =3040

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 2 4 (𝑛+1) 2 WB 12 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 2 4 3 𝑛− 2 3 − 3𝑛 2 𝑛 3 + 1 3 = 3𝑛 2 4 3 𝑛− 2 3 − 𝑛 3 − 1 3 = 3𝑛 2 𝑛−1 QED

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 2 4 (𝑛+1) 2 WB 13 a) show that 𝑟=1 𝑛 2 𝑟 − 𝑟=1 𝑛 𝑟 = 𝑛( 𝑛 3 −1) 2 b) Hence evaluate 𝑟=10 81 𝑟 𝑎) 𝑟=1 𝑛 2 𝑟 − 𝑟=1 𝑛 𝑟 = 𝑛 2 2 𝑛 2 +1 − 𝑛 2 (𝑛+1) = 𝑛 2 𝑛 3 +𝑛 −𝑛 −1 = 𝑛( 𝑛 3 −1) 2 𝑏) 𝑟=10 81 𝑟 = 1 81 𝑟 − 1 9 𝑟 = 9( 9 3 −1) 2 − 3( 3 3 −1) 2 =3237

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

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

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 –

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