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# 3.E: Number Patterns (Exercises)

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#### Exercise $$\PageIndex{1}$$: Hexagonal numbers (cornered)

Consider the hexagonal numbers are the sequence $$1,6,15, 28, \cdots.$$ Predict the n th term. Explain your prediction

#### Exercise $$\PageIndex{2}$$: Finite sum

For each of the following, find the sum and explain your reasoning. Please do not use any formula.

1.  $$1+3+5+7+9+\cdots +197+199$$
2. $$1+\frac{1}{2}+\frac{1}{4}+\cdots +\frac{1}{2^{16}}+\frac{1}{2^{17}}$$

#### Exercise $$\PageIndex{3}$$: Proof by induction

Consider the sequence $$4,10,16, 22, 28,,\dots$$, assume that the pattern continues.

1. Show that the $$n^{th}$$ term of this sequence can be expressed as $$6n-2$$.
2. Prove by using induction for all integers $$n \geq 1, 4+10+16+\dots+(6n-2)=n(3n+1)$$

#### Exercise $$\PageIndex{4}$$: Proof by induction

Consider the sequence $$3,11,19, 27, 35,\dots$$,  assume that the pattern continues.

1. Show that the $$n^{th}$$ term of this sequence can be expressed as $$8n-5$$.
2. Prove by using induction for all integers $$n\geq 1, 3+11+19 \dots + (8n-5)=4n^2-n.$$

#### Exercise $$\PageIndex{5}$$: Tribonacci

Let's start with the numbers $$0,0,1,$$ and generate future numbers in our sequence by adding up the previous three numbers. Write out the first $$15$$ terms in this sequence, starting with the first $$1$$.

#### Exercise $$\PageIndex{6}$$: Proof by induction

The sequence $$b_0,b_1,b_2....$$ is defined as follows: $$b_0=1,b_1=3,b_2=5,$$ and for any integer $$n \geq 3, \, b_n=3b_{n-2}+2b_{n-3}.$$

1.  Find $$b_3,b_4,b_5$$ and $$b_6$$.
2.  Prove that $$b_n < 2^{n+1}$$ for all integers $$n \geq 1.$$

#### Exercise $$\PageIndex{7}$$: Quadratic Sequence

Find the $$n^{th}$$ term of the sequence $$5,10,17, 26, 37, \cdots$$, assume that the pattern continues.

#### Exercise $$\PageIndex{8}$$: Proof by induction

Prove by using induction: for all integers $$n\geq 1, \, 1+4+7 \dots + (3n-2)=\frac{n(3n-1)}{2}.$$

Predict  $$n^{th}$$ term of the sequence $$\frac{2}{3},\frac{3}{4}, \frac{4}{5}\cdots\, assume that the pattern continues. Explain your prediction. #### Exercise \(\PageIndex{10}$$: Recognising sequence

Consider the sequence $$t_1=1, t_2=3+5, t_3=7+9+11, \cdots$$. Predict the n th term. Justify your prediction.

#### Exercise $$\PageIndex{11}$$: Proof by induction

Show that the perimeter of the design by joining  $$n$$ hexagons in a row is  $$8n+4$$ cm.

#### Exercise $$\PageIndex{13}$$: Pentagonal Numbers  (cornered)

Find the $$n^{th}$$ term of the sequence $$1,5,12, 22, \cdots$$,assume that the pattern continues.

#### Exercise $$\PageIndex{14}$$: Square Pyramidal numbers

Find the $$n^{th}$$ term of the sequence  $$1,5,14,30 \cdots$$., assume that the pattern continues.