Question 6

We will find each term by adding the three previous terms.

Step 1 — Find T₄

We are given the first three terms. We use the recurrence relation for $n=4$. $T_n = T_{n-1} + T_{n-2} + T_{n-3}$

$T_4 = T_3 + T_2 + T_1$

$= 4 + 2 + 1$

$\boxed{T_4 = 7}$

Step 2 — Find T₅

We use the recurrence relation for $n=5$. We use the value of $T_4 = \textbf{7}$.

$T_5 = T_4 + T_3 + T_2$

$= 7 + 4 + 2$

$\boxed{T_5 = 13}$

Step 3 — Find T₆

We use the recurrence relation for $n=6$. We use the value of $T_5 = \textbf{13}$.

$T_6 = T_5 + T_4 + T_3$

$= 13 + 7 + 4$

$\boxed{T_6 = 24}$

Step 4 — Find T₇

We use the recurrence relation for $n=7$. We use the value of $T_6 = \textbf{24}$.

$T_7 = T_6 + T_5 + T_4$

$= 24 + 13 + 7$

$\boxed{T_7 = 44}$

Step 5 — Find T₈

We use the recurrence relation for $n=8$. We use the value of $T_7 = \textbf{44}$.

$T_8 = T_7 + T_6 + T_5$

$= 44 + 24 + 13$

$\boxed{T_8 = 81}$

Answer

$T_4 = 7$ $T_5 = 13$ $T_6 = 24$ $T_7 = 44$ $T_8 = 81$