TOPIC 6: SEQUENCES & SERIES I (AP & GP)

Question 1

How to describe the behaviour of a sequence?

Answer 1

1. Increasing / decreasing / alternating
2. Divergent / Convergent to x

Question 2

[General] uₙ

Answer 2

uₙ = Sₙ - Sₙ₋₁

Question 3

[AP] nth term

Answer 3

uₙ = a + (n-1)d

Question 4

[AP] Sₙ

Answer 4

Sₙ = n/2 [a + l]
Sₙ = n/2 [2a + (n-1)d]

Question 5

[AP] Prove sequence is an AP

Answer 5

Show that
uₙ - uₙ₋₁ OR uₙ₊₁ - uₙ
is a constant independent of n

Question 6

[GP] nth term

Answer 6

uₙ = arⁿ⁻¹

Question 7

[GP] Sₙ

Answer 7

Sₙ = a(1-rⁿ) / 1-r
Sₙ = a(rⁿ-1) / r-1

Question 8

[GP] Proving sequence is a GP

Answer 8

Show that
uₙ/uₙ₋₁ OR uₙ₊₁ / uₙ
is a constant independent of n

Question 9

2 ways to find Sum to Infinity

Answer 9

1. As n –> ∞, Sₙ –> X. So, sum to infinity is X
2. Since |r| < 1, the sum to infinity is a/(1-r) = X

Question 10

Standard Steps for Compound Interest Qn

Answer 10

Formulate the following equations

200(1.05)ⁿ + 200(1.05)ⁿ⁻¹ + … + 200(1.05)

Factorise from backwards:
200(1.05)[1 + 1.05 + … + 1.05ⁿ⁻² + 1.05ⁿ⁻¹]

Sₙ (GP Formula):
a (rⁿ-1) / (r-1)
200(1.05) (1.05ⁿ - 1) / (1.05-1)

Solve to get answer