Question 4
Given: $g(n)=n^{3}+5 n$ and $h(n)=3 n+3$
Find: $h(g(n+1))$
$4 n^{3}+12 n^{2}-3 n+14$
$3 n^{3}+9 n^{2}+24 n+21$
$6 n^{2}-4 n+6$
There is no correct answer given.
$-9 n^{3}-24 n+18$
Answer
So, the final answer is \(\boxed{3 n^{3}+9 n^{2}+24 n+21}\).
Step 1 ：Given the functions $g(n)=n^{3}+5 n$ and $h(n)=3 n+3$, we are asked to find the value of $h(g(n+1))$.
Step 2 ：First, we need to find the value of $g(n+1)$.
Step 3 ：Substituting $n+1$ into $g(n)$, we get $g(n+1) = (n+1)^{3}+5(n+1)$.
Step 4 ：Next, we substitute this value into $h(n)$ to get $h(g(n+1)) = 3g(n+1) + 3$.
Step 5 ：Substituting $g(n+1)$ into this equation, we get $h(g(n+1)) = 3((n+1)^{3}+5(n+1)) + 3$.
Step 6 ：Expanding this equation, we get $h(g(n+1)) = 3n^{3} + 9n^{2} + 24n + 21$.
Step 7 ：So, the final answer is \(\boxed{3 n^{3}+9 n^{2}+24 n+21}\).
