Algebra 2 Arithmetic and Geometric Sequences and Series Worksheet Solutions (PDF)

Algebra 2 math worksheet solutions: Arithmetic and Geometric Sequences and Series – Maths PDF

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This algebra 2 sequences series answers for Algebra 2 covers arithmetic and geometric sequences and series (Common Core F-BF.A.2, A-SSE.B.4, F-IF.A.3), with a free PDF to print.

Every worksheet problem on arithmetic and geometric sequences and series is solved here, one step at a time. Compare your method as well as your final answer, because the steps show where mixing up the arithmetic and geometric formulas can go wrong. Final answers appear in bold so you can find them quickly when you only need a check.

Algebra 2 · Common Core State Standards: F-BF.A.2, A-SSE.B.4, F-IF.A.3

Algebra 2 sequences series answers: exercises 1 to 10

Exercise 1 – List the terms

Substitute $n = 1, 2, 3, 4, 5$: $1, 4, 7, 10, 13$.

1, 4, 7, 10, 13

Exercise 2 – Recursive formula

Start at 5 and add 4 each time.

5, 9, 13, 17, 21

Exercise 3 – Arithmetic or geometric?

Completed table.

$6 \div 3 = 2$, $12 \div 6 = 2$: geometric. $17 – 20 = -3$, $14 – 17 = -3$: arithmetic. Ratios 3, 3, 3: geometric.

Geometric, arithmetic, geometric

Exercise 4 – Explicit arithmetic formula

$d = 4$, $a_{1} = 7$: $a_{n} = 7 + (n – 1)4 = 4n + 3$.

$a_{20} = 4(20) + 3 = 83$.

$a_{n} = 4n + 3$; $a_{20} = 83$

Exercise 5 – Find a term

$a_{15} = -4 + 14(3) = -4 + 42 = 38$.

Exercise 6 – Find the position

$a_{n} = 2 + (n – 1)3 = 3n – 1$. Set $3n – 1 = 62$: $n = 21$.

The 21st term

Exercise 7 – Geometric formula

$r = 2$: $a_{n} = 3 \cdot 2^{n-1}$. $a_{8} = 3 \cdot 2^{7} = 3 \cdot 128 = 384$.

$a_{8} = 384$

Exercise 8 – Geometric with a fraction

A geometric sequence with ratio 1/2: each bar is half the one above.

$a_{6} = 80\left(\frac{1}{2}\right)^{5} = \frac{80}{32} = 2.5$.

2.5

Exercise 9 – Recursive geometric

Multiply by 3 each time: 2, 6, 18, 54.

2, 6, 18, 54

Exercise 10 – Gauss's sum

Pairing the sequence with its reverse: every column sums to 11.

$S_{100} = \frac{100(1 + 100)}{2} = 50 \cdot 101$.

5,050

Exercises 11 and up

Exercise 11 – Arithmetic series

$a_{20} = 5 + 19(3) = 62$.

$S_{20} = \frac{20(5 + 62)}{2} = 10 \cdot 67 = 670$.

670

Exercise 12 – Theater seats

$a_{15} = 12 + 14(2) = 40$.

$S_{15} = \frac{15(12 + 40)}{2} = 15 \cdot 26 = 390$.

390 seats

Exercise 13 – Geometric series

$S_{8} = 3\cdot\frac{1 – 2^{8}}{1 – 2} = 3 \cdot \frac{-255}{-1} = 765$.

765

Exercise 14 – Halving series

$r = \frac{1}{2}$. $S_{7} = 64\cdot\frac{1 – (1/2)^{7}}{1 – 1/2} = 64 \cdot \frac{127/128}{1/2} = 127$.

127

Exercise 15 – Doubling pennies

$S_{30} = 1\cdot\frac{1 – 2^{30}}{1 – 2} = 2^{30} – 1 = 1{,}073{,}741{,}823$ cents.

Divide by 100 to get dollars.

$10,737,418.23

Exercise 16 – Savings plan

Geometric series with $a_{1} = 200$, $r = 1.05$, $n = 10$.

$S_{10} = 200\cdot\frac{1 – 1.05^{10}}{1 – 1.05} = 200\cdot\frac{1.62889 – 1}{0.05} \approx 2,515.58$.

About $2,515.58

Exercise 17 – Triangular stack

A stack of logs with 5 in the bottom row.

The sum $1 + 2 + … + 20$ has $n = 20$, $a_{1} = 1$, $a_{20} = 20$.

$S_{20} = \frac{20(1 + 20)}{2} = 210$.

210 logs

Exercise 18 – Fibonacci

Add the two previous terms: 1, 1, 2, 3, 5, 8, 13, 21.

The differences 0, 1, 1, 2 and the ratios 1, 2, 1.5 are not constant.

1, 1, 2, 3, 5, 8, 13, 21; neither

Exercise 19 – Find the error

The differences are $2, 4, 8$, which are not constant. The ratios are all 2, so the sequence is geometric.

It is geometric with r = 2

Exercise 20 – Two terms given

$a_{8} – a_{3} = 5d = 25$, so $d = 5$.

$a_{1} = 11 – 2(5) = 1$. $a_{n} = 1 + 5(n – 1) = 5n – 4$.

$d = 5$, $a_{1} = 1$, $a_{n} = 5n – 4$

Key facts behind this algebra 2 sequences series answers

Key facts

A sequence is a function whose domain is the positive integers.
Arithmetic: a_n = a_1 + (n – 1)d; geometric: a_n = a_1·r^(n-1).
Recursive formulas build each term from the previous one.
Arithmetic sum: S_n = n(a_1 + a_n)/2.
Geometric sum: S_n = a_1(1 – r^n)/(1 – r), r ≠ 1.

How to use this algebra 2 sequences series answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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