You can try to show this by induction:
• According to the given closed form, we have
, which agrees with the initial value <em>S</em>₁ = 1.
• Assume the closed form is correct for all <em>n</em> up to <em>n</em> = <em>k</em>. In particular, we assume

and

We want to then use this assumption to show the closed form is correct for <em>n</em> = <em>k</em> + 1, or

From the given recurrence, we know

so that






which is what we needed. QED
Answer:
I think its better than other types of math.
Step-by-step explanation:
Answer:

b = (T - a - c - d) / 3
Step-by-step explanation:
Let T be the total number of points required to advance.
a, c and d are points scored in the local matches, and b is the number of points scored in the district match. If b is worth 3 times as much as the other matches, the total number of points is given by:

Isolate b in order to find out how many points they need in the district match:

They need to score (T - a - c - d)/3, in the district match in order to win.
Answer:
y" = csc(x)[9cot²(x) - csc²(x)]
Step-by-step explanation:
Step 1: Define
y = 9csc(x)
Step 2: Find 1st derivative
y' = -9csc(x)cot(x)
Step 3: Find 2nd derivative
y" = 9csc(x)cot(x)cot(x) + -csc(x)csc²(x)
y" = 9csc(x)cot²(x) - csc³(x)
y" = csc(x)[9cot²(x) - csc²(x)]