Number patterns

Advanced Math 7 · Unit 9 · sequences-patterns-7 · Teacher edition

sequences-patterns-7

Name ______________________________Date ______________

1.
What is the $10$ th term of the arithmetic sequence $4, 11, 18, 25, \ldots$ ?
Answer: ______________
2.
Write a formula for the $n$ th term $T_n$ of the sequence $7, 12, 17, 22, \ldots$ in terms of $n$ . Give it in the form $an + b$ .
Answer: ______________
3.
A pattern of dots grows as shown. Each figure is a triangle of dots with one more row than the figure before.

Figure 1 2 3 4
Dots 3 6 10 15

How many dots are in Figure $8$ ?
Answer: ______________
4.
Which formula gives the $n$ th term of $3, 7, 11, 15, \ldots$ ?
1. (A)
  $T_n = 4n + 3$
2. (B)
  $T_n = 3n + 4$
3. (C)
  $T_n = n + 4$
4. (D)
  $T_n = 4n - 1$
5.
How many terms of the arithmetic sequence $7, 10, 13, 16, \ldots$ are less than $100$ ?
1. (A)
  $30$
2. (B)
  $31$
3. (C)
  $32$
4. (D)
  $33$
5. (E)
  $34$

Figure	1	2	3	4
Dots	3	6	10	15

Answer key — Number patterns

1.
67sequences-patterns-7-01
$T_{10} = 4 + (10 - 1) \times 7 = 4 + 63 = \mathbf{67}$ . (Adding $d$ ten times, $4 + 70 = 74$ , gives the $11$ th term.)
2.
$5n+2$ sequences-patterns-7-02
$T_n = 7 + 5(n - 1) = 7 + 5n - 5 = \mathbf{5n + 2}$ . Check: $n = 1 \to 7$ , $n = 2 \to 12$ , $n = 3 \to 17$ . ✓ ( $n + 5$ is not a formula for $T_n$ ; it only restates the step.)
3.
45sequences-patterns-7-03
Figure $8$ is the $9$ th triangular number: $\frac{9 \times 10}{2} = \mathbf{45}$ . By differences: $15 + 6 = 21$ , $+7 = 28$ , $+8 = 36$ , $+9 = 45$ . ✓
4.
(D)
$T_n = 4n - 1$
sequences-patterns-7-04
$d = 4$ gives $T_n = 4n + b$ , and $T_1 = 3$ forces $b = -1$ . So $\mathbf{T_n = 4n - 1}$ : $3, 7, 11, 15$ . ✓ ( $n + 4$ gives $5, 6, 7$ — the “add the difference” trap; $4n + 3$ uses $a + dn$ and gives $7, 11, 15$ , one term early.)
5.
(B)
$31$
sequences-patterns-7-05
$3n + 4 < 100 \Rightarrow n < 32$ , so $n = 1, 2, \ldots, 31$ : $\mathbf{31}$ terms. Check: $T_{31} = 97 < 100$ and $T_{32} = 100$ , which is not less than $100$ . ( $32$ counts the term equal to $100$ .)