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Mathematics LibreTexts

3.E: Number Patterns (Exercises)

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

By Onmywaybackhome (Own work) [Public domain], via Wikimedia Commons

 

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 \( 4,10,16,\dots\)

  1. Show that the \(n^{th}\) term of this sequence is \(6n-2\).
  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 \( 3,11,19,\dots\)

  1. Show that the \(n^{th}\) term of this sequence is \(8n-5\).
  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 \(5,10,17, 26, 37, \cdots\)

Exercise \(\PageIndex{8}\): Proof by induction

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

Exercise \(\PageIndex{9}\): Recognising sequence

 Find the \(n^{th}\) term of the sequence \(\frac{2}{3},\frac{3}{4}.\cdots\).

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

An elementary-level math problem (Math Focus 6, page 7, activity 4) is as follows: "Clara made a design by joining hexagons in a row. Each side of each hexagon is 2 cm. Predict the perimeter of Clara's design. Explain your prediction."

We have concluded that the perimeter of the design by joining n hexagons in a row is 8n+4 cm. Prove this result by induction.

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

Image used with permission (Public Domain; Timrem)

Find the \(n^{th}\) term of the sequence \(1,5,12, 22, \cdots\).

Exercise \(\PageIndex{14}\): Square Pyramidal numbers

By David Eppstein at English Wikipedia (Transferred from en.wikipedia to Commons.) [Public domain], via Wikimedia Commons

Find the \(n^{th}\) term of the sequence  \(1,5,14,30 \cdots\).