Find the first five terms of the sequence in which the term is given by, for : (i) (ii) (iii)

Understand the Question The term of a sequence, denoted as , is a general algebraic formula that defines every term in the sequence based on its position . To find the first five terms of each sequence, substitute the natural numbers sequentially into the given formula for . Part (i) Step 1 · Calculate the First Five Terms Substitute into : For : For : For : For : For : Answer > (i) Part (ii) Step 1 · Calculate the First Five Terms Substitute into : For : For : For : For : For : Answer > (ii) Part (iii) Step 1 · Calculate the First Five Terms Substitute into : For : For : For : For : For : Answer > (iii) Common Mistakes Starting Index: Sequences start with (since ). Starting at will yield incorrect initial terms. Sign Errors in Subtraction: In part (ii), be careful with the negative signs when computing (e.g. , not ). Order of Operations: In part (iii), ensure powers and multiplications are evaluated before addition and subtraction ().

Predicting What Comes Next: Sequences and Progressions | Exercise 8.1

Question 1

Find the first five terms of the sequence in which the $n^{\text{th}}$ term is given by, for $n \ge 1$ :

(i) $t_n = 3n - 4$

(ii) $t_n = 2 - 5n$

(iii) $t_n = n^2 - 2n + 3$

Solution

Understand the Question

The $n^{\text{th}}$ term of a sequence, denoted as $t_n$ , is a general algebraic formula that defines every term in the sequence based on its position $n$ .
To find the first five terms of each sequence, substitute the natural numbers $n = 1, 2, 3, 4, 5$ sequentially into the given formula for $t_n$ .

(i) $t_n = 3n - 4$

Step 1 · Calculate the First Five Terms

Substitute $n = 1, 2, 3, 4, 5$ into $t_n = 3n - 4$ :

For $n = 1$ :

\begin{aligned} t_1 &= 3(1) - 4 \\ &= 3 - 4 \\ &= -1 \end{aligned}

For $n = 2$ :

\begin{aligned} t_2 &= 3(2) - 4 \\ &= 6 - 4 \\ &= 2 \end{aligned}

For $n = 3$ :

\begin{aligned} t_3 &= 3(3) - 4 \\ &= 9 - 4 \\ &= 5 \end{aligned}

For $n = 4$ :

\begin{aligned} t_4 &= 3(4) - 4 \\ &= 12 - 4 \\ &= 8 \end{aligned}

For $n = 5$ :

\begin{aligned} t_5 &= 3(5) - 4 \\ &= 15 - 4 \\ &= 11 \end{aligned}

Answer

(i) $-1, 2, 5, 8, 11$

(ii) $t_n = 2 - 5n$

Step 1 · Calculate the First Five Terms

Substitute $n = 1, 2, 3, 4, 5$ into $t_n = 2 - 5n$ :

For $n = 1$ :

\begin{aligned} t_1 &= 2 - 5(1) \\ &= 2 - 5 \\ &= -3 \end{aligned}

For $n = 2$ :

\begin{aligned} t_2 &= 2 - 5(2) \\ &= 2 - 10 \\ &= -8 \end{aligned}

For $n = 3$ :

\begin{aligned} t_3 &= 2 - 5(3) \\ &= 2 - 15 \\ &= -13 \end{aligned}

For $n = 4$ :

\begin{aligned} t_4 &= 2 - 5(4) \\ &= 2 - 20 \\ &= -18 \end{aligned}

For $n = 5$ :

\begin{aligned} t_5 &= 2 - 5(5) \\ &= 2 - 25 \\ &= -23 \end{aligned}

Answer

(ii) $-3, -8, -13, -18, -23$

(iii) $t_n = n^2 - 2n + 3$

Step 1 · Calculate the First Five Terms

Substitute $n = 1, 2, 3, 4, 5$ into $t_n = n^2 - 2n + 3$ :

For $n = 1$ :

\begin{aligned} t_1 &= 1^2 - 2(1) + 3 \\ &= 1 - 2 + 3 \\ &= 2 \end{aligned}

For $n = 2$ :

\begin{aligned} t_2 &= 2^2 - 2(2) + 3 \\ &= 4 - 4 + 3 \\ &= 3 \end{aligned}

For $n = 3$ :

\begin{aligned} t_3 &= 3^2 - 2(3) + 3 \\ &= 9 - 6 + 3 \\ &= 6 \end{aligned}

For $n = 4$ :

\begin{aligned} t_4 &= 4^2 - 2(4) + 3 \\ &= 16 - 8 + 3 \\ &= 11 \end{aligned}

For $n = 5$ :

\begin{aligned} t_5 &= 5^2 - 2(5) + 3 \\ &= 25 - 10 + 3 \\ &= 18 \end{aligned}

Answer

(iii) $2, 3, 6, 11, 18$

Common Mistakes

Starting Index: Sequences start with $n = 1$ (since $n \ge 1$ ). Starting at $n = 0$ will yield incorrect initial terms.
Sign Errors in Subtraction: In part (ii), be careful with the negative signs when computing $2 - 5n$ (e.g. $2 - 10 = -8$ , not $+8$ ).
Order of Operations: In part (iii), ensure powers and multiplications are evaluated before addition and subtraction ( $n^2 - 2n + 3$ ).

Question 1

More questions in Exercise 8.1