Precalculus Sequences and Series Worksheet Solutions (PDF)

Precalculus math worksheet solutions: Sequences and Series – Maths PDF

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

Use this key to check each sequence and sum. Solutions identify the pattern, show the formula used and give the final answer in bold.

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

Precalculus sequences series answers: exercises 1 to 10

Exercise 1 – List terms

$a_{1}=5$, $a_{2}=8$, $a_{3}=11$, $a_{4}=14$.

$5,\ 8,\ 11,\ 14$

Exercise 2 – Explicit arithmetic formula

Constant difference 4.

$d=4$ and $a_{1}=7$, so $a_{n}=7+4(n-1)=4n+3$. $a_{20}=4(20)+3=83$.

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

Exercise 3 – Formula from two terms

$5d=25-10=15$, so $d=3$. $a_{1}=10-2(3)=4$. Then $a_{n}=4+3(n-1)=3n+1$.

$a_{n}=3n+1$

Exercise 4 – Geometric formula

Common ratio 2.

$r=2$, $a_{1}=3$. $a_{10}=3\cdot2^{9}=3\cdot512=1536$.

$a_{n}=3\cdot2^{n-1}$; $a_{10}=1536$

Exercise 5 – Fractional ratio

$r=\frac{27}{81}=\frac13$. $a_{6}=81\left(\frac13\right)^{5}=\frac{81}{243}=\frac13$.

$r=\frac13$; $a_{6}=\frac13$

Exercise 6 – Recursive sequence

$a_{2}=3(2)-1=5$. $a_{3}=3(5)-1=14$. $a_{4}=3(14)-1=41$.

$5,\ 14,\ 41$

Exercise 7 – Recursive to explicit

It is arithmetic with $d=4$: $a_{n}=5+4(n-1)$.

$a_{n}=4n+1$

Exercise 8 – Sum of the first 100 integers

$S_{100}=\frac{100}{2}(1+100)=50\cdot101$.

5050

Exercise 9 – Arithmetic series

$a_{30}=5+29(3)=92$. $S_{30}=\frac{30}{2}(5+92)=15\cdot97$.

$1455$

Exercise 10 – Sum of 20 terms

$a_{20}=83$. $S_{20}=\frac{20}{2}(7+83)=10\cdot90$.

900

Exercises 11 and up

Exercise 11 – Sigma notation

$3+5+7+9+11$.

Exercise 12 – Sigma with an arithmetic formula

The first term is $1$ and the last term is $3(40)-2=118$. $S_{40}=\frac{40}{2}(1+118)=20\cdot119$.

2380

Exercise 13 – Geometric series

$a_{1}=2$, $r=3$, $n=6$. $S_{6}=2\cdot\frac{1-3^{6}}{1-3}=2\cdot\frac{-728}{-2}$.

728

Exercise 14 – Partial sum of a convergent series

$a_{1}=1$, $r=\frac12$. $S_{8}=\frac{1-\left(\frac12\right)^{8}}{1-\frac12}=\frac{255/256}{1/2}=\frac{255}{128}\approx1.992$.

$\frac{255}{128}$

Exercise 15 – Infinite geometric series

The partial sums approach 16.

$r=\frac12$, so $|r|< 1$. $S=\frac{8}{1-\frac12}=16$.

Exercise 16 – Sigma with infinity

$a_{1}=3$ and $r=0.4$. $S=\frac{3}{1-0.4}=5$.

Exercise 17 – Converge or diverge?

(a) $r=3$, $|r|\ge1$: diverges. (b) $r=-\frac12$: converges; $S=\frac{5}{1+\frac12}=\frac{10}{3}$.

(a) no sum; (b) $\frac{10}{3}$

Exercise 18 – Repeating decimal

$0.36+0.0036+\text{ … }$ has $a_{1}=0.36$ and $r=0.01$. $S=\frac{0.36}{0.99}=\frac{36}{99}=\frac{4}{11}$.

$\frac{4}{11}$

Exercise 19 – Salary

$a_{10}=42000+2000(9)=60000$. $S_{10}=\frac{10}{2}(42000+60000)=510{,}000$.

$\$60{,}000$; $\$510{,}000$

Exercise 20 – Bouncing ball

Each bounce is up and down.

The first bounce reaches $6$ ft, and the heights are $6,3.6,2.16,\text{ … }$ with ratio $0.6$. Sum of the up-heights: $\frac{6}{1-0.6}=15$. Each is traveled up and down: $2\cdot15=30$. Add the first drop: $10+30$.

40 ft

Exercise 21 – Doubling pennies

Geometric with $a_{1}=1$, $r=2$, $n=20$: $S_{20}=\frac{2^{20}-1}{2-1}=1{,}048{,}575$ cents $=\$10{,}485.75$.

$\$10{,}485.75$

Key facts behind this precalculus sequences series answers

Key facts

Arithmetic: a_n = a_1 + (n – 1)d; sum S_n = (n/2)(a_1 + a_n).
Geometric: a_n = a_1 r^(n-1); sum S_n = a_1(1 – r^n)/(1 – r).
Sigma notation adds the terms from the lower index to the upper index.
An infinite geometric series converges only if |r| < 1, and then S = a_1/(1 - r).
A recursive formula gives each term from the one before it.

How to use this precalculus 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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