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

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Answer

So, the final answer is $3 n^{3} + 9 n^{2} + 24 n + 21$ .

Steps

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 $h (g (n + 1)) = 3 g (n + 1) + 3$ .

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 $h (g (n + 1)) = 3 n^{3} + 9 n^{2} + 24 n + 21$ .

Step 7 ：So, the final answer is $3 n^{3} + 9 n^{2} + 24 n + 21$ .