Predicting What Comes Next: Exploring Sequences and Progressions

Can you describe the pattern in each of the above sequences? Can you predict the next few numbers in these sequences?

Consider the sequence 1, 4, 7, 10, 13, … Can you predict the next four terms? Can you derive the first 10 terms of the sequence obtained by adding all the terms up to a given term of this sequence? Hint: The first term is 1. The second term is 1 + 4 = 5, the third term is 1 + 4 + 7 = 12, and so on.

Can you write $t_5$, $t_6$, $t_7$ and $t_8$ for the sequence of triangular numbers?

Can you think of any other kinds of sequences? List out five different types of sequences and discuss their properties with your friends.

Why is it useful to have an explicit formula for the $n^{th}$ term of a sequence?

Using the explicit rule $u_n = 2n - 1$, find the $53^{rd}$ term, the $108^{th}$ term, and the $1170^{th}$ term of the odd number sequence.

Can you find the rule describing the $n^{th}$ term of the sequence of square numbers?

Consider the expression $t_n = 3n - 7$. (i) Find its first, second, third, $12^{th}$, $18^{th}$ and $50^{th}$ terms. (ii) Which term of the sequence is 332? (iii) Is 557 a term of this sequence? Why or why not?

Find the first five terms of the sequence in which the $n^{th}$ term is given by (i) $t_n = 3n - 4$, (ii) $t_n = 2 - 5n$, and (iii) $t_n = n^2 - 2n + 3$ for $n \geq 1$.

Find the $10^{th}$ and $15^{th}$ terms of the sequence $t_n = 5n - 3$ for $n \geq 1$.

Determine whether 97 and 172 are terms of the sequence $t_n = 5n - 3$ for $n \geq 1$.

Which term of the sequence $t_n = 5n - 3$ for $n \geq 1$ is 607?

A sequence is given by the recursive rule $t_1 = -5$, $t_{n+1} = t_n + 3$ for $n \geq 1$. Find the first five terms of the sequence. Is 52 a term of this sequence? If so, which term is it?

Let $T_1 = 1$, $T_2 = 2$, $T_3 = 4$, and $T_n = T_{n-1} + T_{n-2} + T_{n-3}$ for $n \geq 4$. Find $T_4$, $T_5$, $T_6$, $T_7$, and $T_8$.

Can you predict the number of squares in Stages 5 and 6 of the sequence? In Stages 10, 11 and 12? In Stage 20? At any stage?

Consider all the sequences we have discussed so far in this chapter. Which ones are arithmetic progressions and which ones are not? Can you justify your claim?

Verify that the following sequences are arithmetic progressions and write their $n^{th}$ terms. What do you observe when you plot the ordered pairs emerging from them? (i) 2, 5, 8, 11, … (ii) −5, −1, 3, 7, …

Using the formula $t_n = a + (n - 1) \times d$, find the $n^{th}$ term of the following arithmetic progressions. (i) $\frac{1}{2}, \frac{5}{2}, \frac{9}{2}, \frac{13}{2}, \ldots$ (ii) 1.5, 3.5, 5.5, 7.5, …

Can the same approach be used to find the sum of $1 + 2 + 3 + \ldots + 100$?

Can you use this formula to find $S_{20}$, $S_{50}$ or $S_{1000}$?