ACT 1 Alg: Identify arithmetic sequences

Welcome to your ACT 1 Alg: Identify arithmetic sequences

$Q_{1}:$ A sequence is recursively defined by $a_{n}=a_{n-1}+\frac{2}{n}$. If $a_{0}=3$, what is the value of $a_{3}$?

$\frac{20}{3}$

$6$

$5$

$\frac{18}{3}$

$Q_{2}:$ Let $f(x)=\sqrt{x^{2}+5}$ find $f\circ f\circ f(1)$?

$4$

$-4$

$3$

$16$

$Q_{3}:$ For next year’s vacation, Cabrera deposited $2,000£$ into a savings account that pays $0.5%$ compounded monthly. In addition to this initial deposit, on the first day of each month, he deposits $200£$ into the account. The amount of money $n$ months after he opened the account can be calculated by the equation, $A_{n}=(1+0.005)A_{n-1}+200$. According to the formula, what will be the amount in Cabrera’s savings account three months after he started it?

$A_{1}=2633.16, A_{2}=2421.05, A_{3}=2210$

$A_{1}=2633.16, A_{2}=2210, A_{3}=2421.05$

$A_{1}=2210, A_{2}=2633.16, A_{3}=2421.05$

$A_{1}=2210, A_{2}=2421.05, A_{3}=2633.16$

$Q_{4}:$ A sequence is recursively defined by $a_{n}=\sqrt{ (a_{n-1})^{2} +2}$. If $a_{0}=\sqrt{2}$, what is the value of $a_{2}$?

$\sqrt{5}$

$\sqrt{8}$

$\sqrt{6}$

$3$

$Q_{5}:$ A sequence is recursively defined by $a_{n+1}=a_{n}-\frac{f(a_{n})}{g(a_{n})}$. If $a_{0}=1$, $f(x)=x^{2}-3x$, and $g(x)=2x-3$, what is the value of $a_{2}$?

$-3$

$-\frac{1}{5}$

$\frac{3}{2}$

$2$

$Q_{6}:$ If $A_{0}$ is the initial amount deposited into a savings account that earns at a fixed rate of $r$ percent per year, and a constant amount of $12b$ is added to the account each year, then amount $A_{n}$ of the savings $n$ years after the initial deposit is made is given by the equation $A_{n}=(1+\frac{r}{100}A_{n-1}+12b$. What is $A_{3}$, the amount you have in the savings three years after you made the initial deposit, if $r = 5 $, $A_{0} = 12000 $, and $b = 400 $?

$23070$

$26048$

$29023,50$

$35274,68$

$Q_{7}:$ The number of gallons, P_{n} , of a pollutant in a lake at the end of each month is given by the recursively defined formula $P_{n}=0.85P_{n-1}+20$. If the initial amount $P_{0}$ of a pollutant in the lake is $400$ gallons, what is $P_{3}$ , the amount of pollutant in the lake at the end of the third month, to the nearest gallon?

$262$

$297$

$273$

$285$

$Q_{8}:$ The tenth term of an arithmetic sequence is $38$, and the second term is $6$. What is the value of the first term of this sequence?

$0$

$14$

$4$

$2$

$Q_{9}:$ What is the $35th$ term of the sequence $-1, 5, -2, 1, 5, 2,...$?

$-2$

$-1$

$2$

$5$

$Q_{10}:$ The first and second terms of a geometric sequence are $a$ and $ab$, in that order. What is the $643$rd term of the sequence?

$(ab)^{642}$

$(ab)^{643}$

$a^{642}b$

$ab^{642}$

$Q_{11}:$ What is the fourth term in the arithmetic sequence $13, 10, 7, …$?

$4$

$9$

$14$

$-7$

$Q_{12}:$ If the pattern of the terms $3\sqrt{3},27,81\sqrt{3},...$ continues, which of the following would be the sixth term of the sequence?

$(3\sqrt{3})^{6}$

$(\sqrt{3})^{6}$

$3^{6}$

$(3\sqrt{3})^{5}$

$Q_{13}:$ Find the $n$th term of the arithmetic sequence $6, 10, 14, 18, 22, . . . $

$a_{n}=4n+2$

$a_{n}=4n-2$

$a_{n}=n+2$

$a_{n}=4n+4$

$Q_{14}:$ The sum of a finite arithmetic sequence is:

$S_{n}=\frac{n}{2}(a_{1}+a_{n})$

$S_{n}=\frac{n}{4}(a_{1}+a_{n})$

$S_{n}=\frac{n}{2}(a_{1}-a_{n})$

$S_{n}=\frac{4n}{2}(a_{1}+a_{n})$

$Q_{15}:$ The $n$th term of an arithmetic sequence can be found using the equation

$a_{n}=dn+3c$

$a_{n}=dn+2c$

$a_{n}=dn+c$

$a_{n}=dn-c$

$Q_{16}:$ Find the sum of the integers from $1$ to $200$.

$20100$

$20101$

$20000$

$40204$

$Q_{17}:$ Find the sum of the terms in the sequence $2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30.$

$200$

$240$

$300$

$250$

$Q_{18}:$ If the $20$th term of an arithmetic sequence is $20$ and the $50$th term is $100$, what is the first term of the sequence?

$-33.33$

$-30.67$

$1$

$2.67$

$Q_{19}:$ If the second term in an arithmetic sequence is $4$, and the tenth term is $15$, what is the first term in the sequence?

$1.18$

$1.27$

$1.38$

$2.63$

$Q_{20}:$ The first term of sequence $I$ is $10$, and each term after the first term is $3$ more than the preceding term. The first term of sequence $II$ is $100$, and each term after the first term is $3$ less than the preceding term. If $x$ is the $16$th term of sequence $I$ and $y$ is the $16$th term of sequence $II$, what is $x + y $?

$110$

$54$

$55$

$200$

$Q_{21}:$ Which of the following sequence is an arithmetic sequence?

$f(n) = k r ^{n} , n \in N$

$f(n) = an + b; n \in N$

$ f(n) = (an + b) k r^{ n}, n \in N$

$f(n) = \frac{1}{ a ( n + \frac{b}{ n} )} ; n \in N$

$Q_{22}:$ Which of the following sequence is NOT an arithmetic sequence?

$10,12,14,16$

$2,8,14,20,26$

$3,15,75,375$

$50,45,40,35,30$

$Q_{23}:$ Trains on your route run every $7$ minutes from $6:30 $A.M. to $10:00 $A.M. You get to the train stop at 7:56 A.M. How long will you have to wait for a train?

$5$ minutes

$10$ minutes

$15$ minutes

$20$ minutes

$Q_{24}:$ Ayah’s exercise plan lasts for $7$ minutes on the first day and increases by $3$ minutes each day.For how long will Ayah exercise on the nineteenth day?

$61$

$74$

$81$

$90$

$Q_{25}:$ This expression $-6,1,8,15,22$ is called

Geometric sequence

Arithmetic sequence

quadratic sequence

special sequence

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