Question

inductive reasoning
the process of arriving at a conclusion based on a set of observations
example: holly notices that every saturday, her neighbor mows his lawn. today is saturday. holly concludes her neighbor will
mow his lawn.
for a,b and c, use inductive reasoning to describe the pattern. then find the 10th term in the pattern.
for exercise 1 - 3, fill in the missing numbers by finding the pattern in the table.
1.
2.
3.

1. how many triangles are formed when you draw all the possible diagonals from just

one vertex of a 35 - gon?

Explanation:

Step1: Analyze the pattern of \(f(n)\) in exercise 1

We observe that \(f(n)\) forms an arithmetic sequence. The common difference \(d = 9 - 3=6\). The formula for an arithmetic sequence is \(f(n)=a+(n - 1)d\), where \(a = 3\) and \(d = 6\). So \(f(n)=3+(n - 1)\times6=6n-3\).

Step2: Calculate \(f(7)\), \(f(8)\), \(f(9)\) and \(f(20)\)

• For \(n = 7\): \(f(7)=6\times7-3=39\)
• For \(n = 8\): \(f(8)=6\times8-3 = 45\)
• For \(n = 9\): \(f(9)=6\times9-3=51\)
• For \(n = 20\): \(f(20)=6\times20-3=117\)

Step3: Analyze the pattern of \(f(n)\) in exercise 2

We observe that \(f(n)\) forms an arithmetic sequence. The common difference \(d=-2 - 1=-3\). The formula for an arithmetic sequence is \(f(n)=a+(n - 1)d\), where \(a = 1\) and \(d=-3\). So \(f(n)=1+(n - 1)\times(-3)=-3n + 4\).

Step4: Calculate \(f(7)\), \(f(8)\), \(f(9)\) and \(f(20)\)

• For \(n = 7\): \(f(7)=-3\times7 + 4=-17\)
• For \(n = 8\): \(f(8)=-3\times8+4=-20\)
• For \(n = 9\): \(f(9)=-3\times9 + 4=-23\)
• For \(n = 20\): \(f(20)=-3\times20+4=-56\)

Step5: Analyze the pattern of \(f(n)\) in exercise 3

We observe that \(f(n)\) forms an arithmetic sequence. The common difference \(d=4-(-4)=8\). The formula for an arithmetic sequence is \(f(n)=a+(n - 1)d\), where \(a=-4\) and \(d = 8\). So \(f(n)=-4+(n - 1)\times8=8n-12\).

Step6: Calculate \(f(7)\), \(f(8)\), \(f(9)\) and \(f(20)\)

• For \(n = 7\): \(f(7)=8\times7-12=44\)
• For \(n = 8\): \(f(8)=8\times8-12=52\)
• For \(n = 9\): \(f(9)=8\times9-12=60\)
• For \(n = 20\): \(f(20)=8\times20-12=148\)

Step7: Analyze the number of triangles formed from one vertex of a polygon

The number of triangles formed by drawing all possible diagonals from one vertex of an \(n\) - gon is \(n - 2\).

Step8: Calculate the number of triangles for a 35 - gon

For \(n = 35\), the number of triangles is \(35-2=33\)

Answer:

1. \(f(7) = 39\), \(f(8)=45\), \(f(9)=51\), \(f(20)=117\)
2. \(f(7)=-17\), \(f(8)=-20\), \(f(9)=-23\), \(f(20)=-56\)
3. \(f(7)=44\), \(f(8)=52\), \(f(9)=60\), \(f(20)=148\)
4. The number of triangles formed from one vertex of a 35 - gon is \(33\)