Question

For each sequence, find the next four terms.

(i) 1, 2, 4, 7, 11, .....

Answer 1

(i) In the given sequence, we have:
t₁ = 1, t₂ = 2, t₃ = 4, t₄ = 7 and t₅ = 11
Thus, we get:
t₁ = 1
t₂ = 2 = 1 + 1 = t_{(2 - 1)} + (2 - 1)
t₃ = 4 = 2 + 2 = t_{(3 - 1)} + (3 - 1)
t₄ = 7 = 4 + 3 = t_{(4 - 1)} + (4 - 1)
t₅ = 11 = 7 + 4 = t_{(5 - 1)} + (5 - 1)
$\Rightarrow$t_n = t_{(n - 1)} + (n - 1)
Now, to find the next four terms, we will take n = 6, 7, 8 and 9.
We get:
t₆ = t_{(6 - 1)} +(6 - 1) = t₅ + 5 = 11 + 5 = 16
_{$$\begin{gathered} t_7=t_{(7-1)}+(7-1)=t_6+6=16+6=22 \\ {t}_8=t_{(8-1)}+(8-1)=t_7+7=22+7=29 \\ {t}_9=t_{(9-1)}+(9-1)=t_8+8=29+8=37 \end{gathered}$$}

Thus, the next four terms are 16, 22, 29 and 37.