Question

Let < a_n > be a sequence. Write the first five terms in each of the following:
(i) a₁ = 1, a_n = a_{n − 1} + 2, n ≥ 2
(ii) a₁ = 1 = a₂, a_n = a_{n − 1} + a_{n − 2}, n > 2
(iii) a₁ = a₂ = 2, a_n = a_{n − 1} − 1, n > 2

Answer 1

(i) a₁ = 1, a_n = a_{n − 1} + 2, n ≥ 2
$$\begin{gathered} a_2=a_1+2=1+2=3 \\ {a}_3=a_2+2=5 \\ {a}_4=a_3+2=7 \\ {a}_5=a_4+2=9 \end{gathered}$$

Hence, the five terms are 1, 3, 5, 7 and 9.

(ii) a₁ = 1 = a₂, a_n = a_{n − 1} + a_{n − 2}, n > 2
$$\begin{gathered} a_3=a_2+a_1=1+1=2 \\ {a}_4=a_3+a_2=2+1=3 \\ {a}_5=a_4+a_3=3+2=5 \end{gathered}$$

Hence, the five terms are 1, 1, 2, 3 and 5.

(iii) a₁ = a₂ = 2, a_n = a_{n − 1} − 1, n > 2
$$\begin{gathered} a_3=a_2-1=2-1=1 \\ {a}_4=a_3-1=1-1=0 \\ {a}_5=a_4-1=0-1=-1 \end{gathered}$$

Hence, the five terms are 2, 2, 1, 0 and $-$1.