Patterns, sequences, and series

High School Math 1 · Unit 13 · arithmetic-geometric-sequences · Teacher edition

arithmetic-geometric-sequences

Name ______________________________Date ______________

1.
An arithmetic sequence begins $7, 11, 15, 19, \dots$ What is its $20$ th term?
Answer: ______________
2.
A sequence begins $-2, 3, 8, 13, \dots$ Write an explicit formula for its $n$ th term. Enter just the expression for $a_n$ in terms of $n$ .
Answer: ______________
3.
A ball is dropped and each bounce rises to $\dfrac{3}{4}$ of the height of the previous bounce. The first bounce reaches $48$ cm. How high, in cm, does the fourth bounce reach?
Answer: ______________
4.
What is the $15$ th term of the arithmetic sequence $50, 44, 38, 32, \dots$ ?
1. (A)
  $-40$
2. (B)
  $-28$
3. (C)
  $-34$
4. (D)
  $134$
5.
The sum of the first $n$ terms of the arithmetic sequence $5, 8, 11, 14, \dots$ is $440$ . What is $n$ ?
1. (A)
  $14$
2. (B)
  $15$
3. (C)
  $16$
4. (D)
  $17$
5. (E)
  $18$

Answer key — Patterns, sequences, and series

1.
83arithmetic-geometric-sequences-01
$a_{20} = 7 + 19 \cdot 4 = 7 + 76 = \mathbf{83}$ . ( $7 + 20 \cdot 4 = 87$ counts one jump too many.)
2.
$5n-7$ arithmetic-geometric-sequences-02
$a_n = -2 + 5(n - 1) = \mathbf{5n - 7}$ . Check: $n = 1$ gives $-2$ , $n = 4$ gives $13$ . ✓
3.
20.25arithmetic-geometric-sequences-03
$g_4 = 48 \cdot \frac{27}{64} = \frac{3 \cdot 27}{4} = \frac{81}{4} = \mathbf{20.25}$ cm. Step by step: $48 \to 36 \to 27 \to 20.25$ .
4.
(C)
$-34$
arithmetic-geometric-sequences-04
$a_{15} = 50 + 14(-6) = 50 - 84 = \mathbf{-34}$ . ( $-40$ uses $15$ jumps, $a_1 + nd$ ; $-28$ uses only $13$ ; $134$ adds $6$ instead of subtracting.)
5.
(C)
$16$
arithmetic-geometric-sequences-05
$n(3n + 7) = 880$ . Try $n = 16$ : $16 \cdot 55 = 880$ . ✓ So $n = \mathbf{16}$ . (Check by the formula: $a_{16} = 50$ and $S_{16} = \frac{16}{2}(5 + 50) = 8 \cdot 55 = 440$ . For $n = 15$ , $15 \cdot 52 = 780 \ne 880$ .)