A lvl H2 Maths: Sequences

The sequence of real numbers b₁, b₂, b₃, ..., is such that b₁ = 4 and b_n = b_n-1 + 2n for all n ∈ Z⁺, n≥2. Using the method of difference, show that b_n = n² + n + 2 for all n ∈ Z⁺.

*************************

Answer:

Using Method of difference,

b_n - b₁
= b_n - b_n-1
+ b_n-1 - b_n-2
+ b_n-2 - b_n-3
+ b_n-3 - b_n-4
+ ..... - .....
+ b₃ - b₂
+ b₂ - b₁

= 2(2) + 2(3) + ... + 2(n-2) + 2(n-1) + 2(n)
= an AP with first term 4, last term 2n, and (n-1) terms
= (n-1)/2 * (4 + 2n)
= (n-1)(n+2)
= n² + n - 2


So,
b_n - 4 = n² + n - 2
b_n = n² + n + 2 (shown)

Related Articles by Categories