GRADE 12 NSC MATHEMATICS EXAMINABLE PROOFS

GRADE 12 NSC MATHEMATICS EXAMINABLE PROOFS

In preparation for the Final Examination learners must ensure that they learn the following proofs:
In Paper 1 (Sum of Arithmetic Series Formula & Sum of the Geometric Series Formula)
In Paper 2 ( Sine Rule, Cosine Rule, Area Rule) & (Six examinable proofs in Euclidean Geometry)

This is a simple 4 page document that contains ALL the proofs…..use it to learn them
Remember this !!! A Geometry Theorem proof will always be tested….and if any other proof comes out that
would mean approximately 10 marks for Theorems….that is approximately 3% of your entire Mathematics
Mark. That could be the difference between passing or failing…or between A and a B symbol. DO NOT
WASTE THE OPPORTUNITY TO GET EASY MARKS!!!


PAPER 1 EXAMINABLE PROOFS


Proof of the Sum of an Arithmetic Series Formula


𝑆𝑛 = 𝑎 + (𝑎 + 𝑑) + (𝑎 + 2𝑑) … … … . . +[𝑎 + (𝑛 − 3)𝑑] + [𝑎 + (𝑛 − 2)𝑑] + [𝑎 + (𝑛 − 1)𝑑] …..(1)

𝑆𝑛 = [𝑎 + (𝑛 − 1)𝑑] + [𝑎 + (𝑛 − 2)𝑑] + [𝑎 + (𝑛 − 3)𝑑] … … … . . +(𝑎 + 2𝑑) + (𝑎 + 𝑑) + 𝑎 …..(2)

(1)+(2)…2𝑆𝑛 = [2𝑎 + (𝑛 − 1)𝑑] + [2𝑎 + (𝑛 − 1)𝑑] + [2𝑎 + (𝑛 − 1)𝑑] + [2𝑎 + (𝑛 − 1)𝑑] … … …

2𝑆𝑛 = 𝑛[2𝑎 + (𝑛 − 1)𝑑]
𝑛
𝑆𝑛 = [2𝑎 + (𝑛 − 1)𝑑]
2



Proof of the Sum of the Geometric Series Formula

𝑆𝑛 = 𝑎 + 𝑎𝑟 + 𝑎𝑟 2 +. … … … … . … + 𝑎𝑟 (𝑛−3) + 𝑎𝑟 (𝑛−2) + 𝑎𝑟 (𝑛−1) ..………….(1)

𝑟𝑆𝑛 = 𝑎𝑟 + 𝑎𝑟 2 + 𝑎𝑟 3 . … … … . +𝑎𝑟 (𝑛−3) + 𝑎𝑟 (𝑛−2) + 𝑎𝑟 (𝑛−1) + 𝑎𝑟 𝑛 ..…..(2)

(2) – (1) 𝑟𝑆𝑛 − 𝑆𝑛 = 𝑎𝑟 𝑛 − 𝑎

𝑆𝑛 (𝑟 − 1) = 𝑎(𝑟 𝑛 − 1)
𝑎(𝑟 𝑛 −1) 𝑎(1−𝑟 𝑛 )
𝑆𝑛 = or if (1) – (2) 𝑆𝑛 =
(𝑟−1) (1−𝑟)




Mr. M. Govender Harry Gwala District 2020