Proof of GP and AP (exclusive)

Question 1

Prove Sn = n/2 (2a + (n-1)d)
(AP)… part 1… whiteboard OR an exclusive item

Answer 1

Sn = a + a+d + a+2d + a+3d + ….. a + (n-1)d
Let a + (n-1)d = L = last term

Question 2

Prove Sn = n/2 (2a + (n-1)d)
(AP)… part 2… whiteboard OR an exclusive item

Answer 2

Sn = a + a+d + a+2d + a+3d + ….. (L-2d) + (L-d) + L
Sn = L + (L-d) + (L-2d) + ….. + a+3d + a+2d + a+d + a
2Sn = (a+L) + (a+L) + (a+L) + (a+L) + ….. (a+L) + (a+L) + (a+L)

Question 3

Prove Sn = n/2 (2a + (n-1)d)
(AP)… part 3… whiteboard OR an exclusive item

Answer 3

Sn = n (a+L)
Sn = n/2 (a+L) = n/2 (a+a+ (n-1)d) = n/2 (2a + (n-1)d)

Question 4

Prove S = a(1 - rⁿ)/1 - r
(GP)… part 1… whiteboard OR another exclusive item

Answer 4

Underline = cross out

S = a + ar + ar² + ar³ + ….. + ar^(n-1)
rS = ar + ar² + ar³ + ar⁴ + …. + ar^(n-1) + arⁿ

Question 5

Prove S = a(1 - rⁿ)/1 - r
(GP)… part 2… whiteboard OR another exclusive item

Answer 5

S - rS = a - arⁿ
S(1-r) = a(1-rⁿ)
S = a(1 - rⁿ)/1 - r