Integers - ACT Math
Card 0 of 1710
Find the sum of the first fifteen terms in an arithmetic sequence whose sixth term is
and whose ninth term is
.
Find the sum of the first fifteen terms in an arithmetic sequence whose sixth term is and whose ninth term is
.
Use the formula _a_n = _a_1 + (n – 1)d
_a_6 = a_1 + 5_d
_a_9 = a_1 + 8_d
Subtracting these equations yields
_a_6 – a_9 = –3_d
–7 – 8 = –3_d_
d = 5
_a_1 = 33
Then use the formula for the series; = –30
Use the formula _a_n = _a_1 + (n – 1)d
_a_6 = a_1 + 5_d
_a_9 = a_1 + 8_d
Subtracting these equations yields
_a_6 – a_9 = –3_d
–7 – 8 = –3_d_
d = 5
_a_1 = 33
Then use the formula for the series; = –30
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If the first day of the year is a Monday, what is the 295th day?
If the first day of the year is a Monday, what is the 295th day?
The 295th day would be the day after the 42nd week has completed. 294 days/7 days a week = 42 weeks. The next day would therefore be a monday.
The 295th day would be the day after the 42nd week has completed. 294 days/7 days a week = 42 weeks. The next day would therefore be a monday.
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If the first two terms of a sequence are
and
, what is the 38th term?
If the first two terms of a sequence are and
, what is the 38th term?
The sequence is multiplied by
each time.
The sequence is multiplied by each time.
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Find the
term of the following sequence:

Find the term of the following sequence:
The formula for finding the
term of an arithmetic sequence is as follows:

where
= the difference between consecutive terms
= the number of terms
Therefore, to find the
term:




The formula for finding the term of an arithmetic sequence is as follows:
where
= the difference between consecutive terms
= the number of terms
Therefore, to find the term:
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What is the
th term in the following series of numbers:
?
What is the th term in the following series of numbers:
?
Notice that between each of these numbers, there is a difference of
. This means that for each element, you will add
. The first element is
or
. The second is
or
, and so forth... Therefore, for the
th element, the value will be
or
.
Notice that between each of these numbers, there is a difference of . This means that for each element, you will add
. The first element is
or
. The second is
or
, and so forth... Therefore, for the
th element, the value will be
or
.
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What is the
rd term of the following sequence:
?
What is the rd term of the following sequence:
?
Notice that between each of these numbers, there is a difference of
; however the first number is
, the second
, and so forth. This means that for each element, you know that the value must be
, where
is that number's place in the sequence. Thus, for the
rd element, you know that the value will be
or
.
Notice that between each of these numbers, there is a difference of ; however the first number is
, the second
, and so forth. This means that for each element, you know that the value must be
, where
is that number's place in the sequence. Thus, for the
rd element, you know that the value will be
or
.
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Tom has twice as many chores as Jeff, who has 3 less than Steve. Steve has 10 chores. How many chores does Tom have?
Tom has twice as many chores as Jeff, who has 3 less than Steve. Steve has 10 chores. How many chores does Tom have?
If Steve has 10 chores, Jeff has 7 chores (3 less than Steve). Tom has twice as many as Steve, so Tom has 14 chores.
If Steve has 10 chores, Jeff has 7 chores (3 less than Steve). Tom has twice as many as Steve, so Tom has 14 chores.
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Your mom made
cookies. You ate
, your brother ate twice as many as you did, and your sister ate
. How many are left?
Your mom made cookies. You ate
, your brother ate twice as many as you did, and your sister ate
. How many are left?
Create an equation for the situation described and solve:



Create an equation for the situation described and solve:
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Your mom gives out
total presents each year to your cousins. If your cousin Sally received
and your cousin Billy received
times as many as Sally, how many did your
cousin, Becky get?
Your mom gives out total presents each year to your cousins. If your cousin Sally received
and your cousin Billy received
times as many as Sally, how many did your
cousin, Becky get?
Create an equation to decribe the above situation and solve.
Total presents given out is represented by
.
Presents given to Sally is represented by
.
Presents given to Billy is represented by
.
Thus our equation becomes,


Create an equation to decribe the above situation and solve.
Total presents given out is represented by .
Presents given to Sally is represented by .
Presents given to Billy is represented by .
Thus our equation becomes,
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Always start by gathering your facts. You know that if
is odd, then neither of those numbers are even. (If either were, then the product would be even.) Now, the rules for subtracting are just like adding. If the difference of two numbers is even, then they must either both be odd or both be even. Thus if
is even and we know that
is odd, then we know that
must be odd. Thus if
is odd, it is also true that
is even, for both
and
are even.
Always start by gathering your facts. You know that if is odd, then neither of those numbers are even. (If either were, then the product would be even.) Now, the rules for subtracting are just like adding. If the difference of two numbers is even, then they must either both be odd or both be even. Thus if
is even and we know that
is odd, then we know that
must be odd. Thus if
is odd, it is also true that
is even, for both
and
are even.
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Which equation can be used to describe the following number line:

Which equation can be used to describe the following number line:

Solve each equation for
, plot on number line.
Solve each equation for , plot on number line.
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On a real number line, what is the midpoint of –4 and 18?
On a real number line, what is the midpoint of –4 and 18?
The integer halfway between –4 and 18 is 7. The number 7 is 11 greater than –4 and 11 less than 18. Another way to determine this would be to find the average of the two numbers: (–4 + 18)/2 = 14/2 = 7.
The integer halfway between –4 and 18 is 7. The number 7 is 11 greater than –4 and 11 less than 18. Another way to determine this would be to find the average of the two numbers: (–4 + 18)/2 = 14/2 = 7.
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Which number below is farthest from –3 on a number line?
Which number below is farthest from –3 on a number line?
The answer here relies on knowledge of how numbers are arranged on a number line. We are looking for the number furthest from –3. The number furthest from –3 can lie either to the left of –3 which would be negative numbers that are more negative, or further from 0 than –3. The number furthest from –3 could also lie to its right on the number line, these numbers include all negative numbers between –3 and 0,0, and all positive numbers. The number that satisifes these conditions and is the furthest from –3 on the number line is –8
The answer here relies on knowledge of how numbers are arranged on a number line. We are looking for the number furthest from –3. The number furthest from –3 can lie either to the left of –3 which would be negative numbers that are more negative, or further from 0 than –3. The number furthest from –3 could also lie to its right on the number line, these numbers include all negative numbers between –3 and 0,0, and all positive numbers. The number that satisifes these conditions and is the furthest from –3 on the number line is –8
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On a number line, what is the distance between –6 and 7?
On a number line, what is the distance between –6 and 7?
To find distance between two points, take the second and subtract the first.
So 7–(–6) = 7 + 6 = 13
To find distance between two points, take the second and subtract the first.
So 7–(–6) = 7 + 6 = 13
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What is the distance on a number line between
and
?
What is the distance on a number line between and
?
The distance between 2 numbers on a number line is the sum of their absolute values.
![[-5] + [12] = 5 + 12 = 17](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/84655/gif.latex)
The distance between 2 numbers on a number line is the sum of their absolute values.
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Add: 
Add:
Simply the signs before solving. A positive sign multiplied with a negative sign will convert the sign to a negative, and a negative multiplied with a negative will convert the sign to a positive.

Simply the signs before solving. A positive sign multiplied with a negative sign will convert the sign to a negative, and a negative multiplied with a negative will convert the sign to a positive.
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Solve: 
Solve:
Starting at
and adding
is the same as subtracting
.
.
Starting at and adding
is the same as subtracting
.
.
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Add 
Add
Starting at
and adding
is the same as subtracting
.
.
Starting at and adding
is the same as subtracting
.
.
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Solve: 
Solve:
Adding
to
is the same as subtracting
from
.
.
Adding to
is the same as subtracting
from
.
.
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If four consecutive odd integers greater than 9 are added together, what is the smallest possible sum of those four integers?
If four consecutive odd integers greater than 9 are added together, what is the smallest possible sum of those four integers?
The 4 consecutive of integers greater than 9 (but not including 9) are 11, 13, 15, 17. Added together, we get 56.
The 4 consecutive of integers greater than 9 (but not including 9) are 11, 13, 15, 17. Added together, we get 56.
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