Porisma

Worked solution

Generate terms and describe a formula

(a) Write down the first four terms of (i) un=1/n2u_n=1/n^2, (ii) un=2n−1u_n=2n-1, for positive integers nn. (b) Write down a general formula for (i) 1,1/2,1/3,1/4,1/51,1/2,1/3,1/4,1/5; (ii) 3,5,7,9,11,…3,5,7,9,11,\ldots; (iii) −1,1,−1,1,−1,…-1,1,-1,1,-1,\ldots.

CJC · Sequences and Series · Self-Practice 1

(a)(i) Reciprocal squares

Answer

1,14,19,1161,\frac14,\frac19,\frac1{16}.

Start at index one

(u1,u2,u3,u4)=(112,122,132,142)\begin{aligned}&(u_1,u_2,u_3,u_4)\\ & =\left(\frac1{1^2},\frac1{2^2},\frac1{3^2},\frac1{4^2}\right)\end{aligned}

These substitutions are authored working; the four results are printed.

How did this part go?

(a)(ii) Odd terms

Answer

1,3,5,71,3,5,7.

Substitute four positive indices

(2(1)−1,2(2)−1,2(3)−1,2(4)−1)=(1,3,5,7)\begin{aligned}&(2(1)-1,2(2)-1,2(3)-1,2(4)-1)\\ & =(1,3,5,7)\end{aligned}

The index is the input, not the preceding term.

How did this part go?

(b)(i) A finite reciprocal sequence

Answer

un=1/nu_n=1/n, n∈{1,2,3,4,5}n\in\{1,2,3,4,5\}.

Match each listed position

u1=1,u2=12,u3=13,u4=14,u5=15u_1=1,\quad u_2=\frac12,\quad u_3=\frac13,\quad u_4=\frac14,\quad u_5=\frac15

Keep the printed finite domain; there is no ellipsis in this ask.

How did this part go?

(b)(ii) Odd terms starting at three

Answer

un=2n+1u_n=2n+1, n∈Z+n\in\mathbb Z^+.

Choose the formula matching the intended pattern

(u1,u2,u3)=(2(1)+1,2(2)+1,2(3)+1)=(3,5,7)\begin{aligned}&(u_1,u_2,u_3)\\ & =(2(1)+1,2(2)+1,2(3)+1)\\ & =(3,5,7)\end{aligned}

This is the printed simple continuation; a finite prefix alone never determines every possible infinite continuation.

How did this part go?

(b)(iii) Alternating signs

Answer

un=(−1)nu_n=(-1)^n, n∈Z+n\in\mathbb Z^+.

Check the first sign

(−1)1=−1,(−1)2=1,(−1)3=−1(-1)^1=-1,\quad(-1)^2=1,\quad(-1)^3=-1

Odd positive indices give −1; even ones give +1.

How did this part go?

Common mistakes

A finite sequence needs its finite index domain. Starting at n=0 would change the first sign and can make a reciprocal formula undefined.

Solution notes

The printed first-four-term answer lists end with ellipses; only four terms are requested.