3.E: Number Patterns (Exercises)

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- 3.4: Finite Difference Calculus
- 4: Basic Concepts of Euclidean Geometry

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

Consider the hexagonal numbers are the sequence $1,6,15, 28,45,66 \cdots.$ Predict the n^th term. Explain your prediction.

Answer: $2n^2-n$ .

Exercise $\PageIndex{2}$ : Finite sum

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

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

Answer

$1+3+5+7+9+···+197+199$

Notice that $1,3,5,7 ,\cdots$ terms of a sequence. This is an Arithmetic Sequence because the difference remains the same between the terms throughout the entire sequence. Hence, $a = 1 \& \, d = 2$ .

Consider,

$S_ n = 1+3+5+7+9+···+197+199$

$S n = 199+197+195+193+191+···+3+1$

By adding we get,

$(2S_n = 200+200+200+200+200+···+200+200$

$2S_n = 100(200)$

$S_n = ((100)/2))(200)$

$S_n = (50)(200)$

$S_n = 10000$

Hence, the sum of the sequence is $10000.$

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

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

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

Answer

Term	First difference
4
10	6
16	6
22	6

Notice that the first difference is constant. Hence the $n^{th}$ term is a linear function.

Let $t_n = a_n + b.$

Then we need to find $a, b$ .

First Equation: Let $n = 1$

$t_1 = 4$

$4 = a(1) + b$

$4 = a + b$

Second Equation: Let $n = 2$

$t_2 = 10$

$10 = a(2) + b$

$10 = 2a + b$

To find $a$ , we use $10 = 2a + b$ and $- 4 = a + b$ . Therfore, $6 = a.$

Now to find $b$ , we use $a = 6$ and $4 = a + b$ ,

$4 = (6) + b$

$4 - 6 = b$

$-2 = b$ .

Hence, $t_n = 6n - 2.$

2. Step 1: Base Step: Show that this statement is true for the smallest value

Check statement is true for n = 1.

L.H.S = 4

R.H.S = n(3n + 1)

= (1)(3(1) + 1)

= (1)(3 + 1)

= (1)(4)

= 4

Hence, the statement is true for n = 1.

Step 2: Induction Assumption:

We shall assume that the statement is true for n = k.

4 + 10 + 16 + . . . + (6k − 2) = k(3k + 1)

Step 3: Induction:

We shall show that the statement is true for n = k + 1.

4 + 10 + 16 + . . . + (6k − 2) + (6 (k + 1) − 2) = (k+1)(3(k + 1) + 1)

Consider, L.H.S = 4 + 10 + 16 + . . . + (6k − 2) + (6 (k + 1) − 2)

= k(3k + 1) + (6 ( k + 1) - 2)

= k (3k + 1) + (6k + 6 - 2)

= k (3k + 1) + (6k + 4)

= 3k 2 + k + 6k + 4

= 3k 2 + 7k + 4

= (k + 1)(3k + 4)

Hence, the statement is true for n = k + 1

Therefore, by induction the statement is true, ∀n ∈ N.

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

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

Show that the $n^{th}$ term of this sequence can be expressed as $8n-5$ .
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 $n \geq 3, \, b_n=3b_{n-2}+2b_{n-3}.$

Find $b_3,b_4,b_5$ and $b_6$ .
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.

Answer: $(n+1)^2+1=n^2+2n+2$

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

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

Answer

Step 1: Base Step: Show that this statement is true for the smallest value

Check statement is true for n = 1.

L.H.S = 1

R.H.S = n(3n−1) / (2)

= (1)(3(1) − 1) / (2)

= (1)(3 − 1) / (2)

= (1)(2) / (2)

= 1

Hence, the statement is true for n = 1.

Step 2: Induction Assumption:

We shall assume that the statement is true for n = k.

1+4+7...+(3k−2)= k(3k−1) / (2)

Step 3: Induction:

We shall show that the statement is true for n = k + 1.

1+4+7...+ (3k - 2) + (3(k + 1) − 2) = (k + 1)(3(k + 1) − 1) / (2)

Consider, L.H.S = k(3k − 1) / (2) + (3 (k + 1) − 2)

= k (3k − 1) / (2) + (3k + 3) − 2)

= k (3k − 1) / (2) + (3k + 1)

= (3k 2 + k ) / (2) + (3k + 1)

= (3k 2 + k + 3k + 1) / (2)

= (3k 2 + 4k + 1) / (2)

= ((k + 1)(3k + 1)) / (2)

Hence, the statement is true for n = k + 1

Therefore, by induction the statement is true, ∀n ∈ N.

Exercise $\PageIndex{9}$ : Recognizing sequence

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.

Answer: $\frac{n}{n+1}$ .

Exercise $\PageIndex{10}$ : Recognizing 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.

Exercise $\PageIndex{15}$ : Difference

Compute the difference of each of the following sequences:

$a_n= n^3$
$a_n=n^{\underline{3}}$
$a_n= \displaystyle {n \choose 3 }$

Answer

$n^2+2n+1$
$3 n^2$
$n \choose 2$ .

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Exercise 3.E.1\PageIndex{1}: Hexagonal numbers (cornered)

Exercise 3.E.2\PageIndex{2}: Finite sum

Exercise 3.E.3\PageIndex{3}: Proof by induction

Exercise 3.E.4\PageIndex{4}: Proof by induction

Exercise 3.E.5\PageIndex{5}: Tribonacci

Exercise 3.E.6\PageIndex{6}: Proof by induction

Exercise 3.E.7\PageIndex{7}: Quadratic Sequence

Exercise 3.E.8\PageIndex{8}: Proof by induction

Exercise 3.E.9\PageIndex{9}: Recognizing sequence

Exercise 3.E.10\PageIndex{10}: Recognizing sequence

Exercise 3.E.11\PageIndex{11}: Proof by induction

Exercise 3.E.13\PageIndex{13}: Pentagonal Numbers (cornered)

Exercise 3.E.14\PageIndex{14}: Square Pyramidal numbers

Exercise 3.E.15\PageIndex{15}: Difference

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

Exercise $\PageIndex{2}$ : Finite sum

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

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

Exercise $\PageIndex{5}$ : Tribonacci

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

Exercise $\PageIndex{7}$ : Quadratic Sequence

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

Exercise $\PageIndex{9}$ : Recognizing sequence

Exercise $\PageIndex{10}$ : Recognizing sequence

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

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

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

Exercise $\PageIndex{15}$ : Difference