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Mathematics LibreTexts

3.E: Number Patterns (Exercises)

  • Page ID
    4900
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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 \( 4,10,16,\dots\)

    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

    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 \(1,5,12, 22, \cdots\).

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

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