Download Series and Sequences - Answer Explanations

Series and Sequences
A series is a progression of numbers that progresses according to a defined pattern. A
sequence refers to the sum of all the terms in a series. For the purposes of the SAT and ACT,
you should be familiar with arithmetic and geometric series and sequences.
Arithmetic Series
An arithmetic series is a series in which a certain number is added to or subtracted from each
term to produce the next term. The number that is added to each term to produce the next
term is known as the common difference. If you subtract the previous term from any term in
the series, the result is equal to the common difference. For instance, 5, 7, 9, 11, 13, 15… is an
arithmetic series with a first term of 5 and a common difference of 2. 10, 7, 4, 1, -2, -5… is an
arithmetic series with a first term of 10 and a common difference of -3.
The terms of an arithmetic series progress as follows: a, a + d, a + 2d, a +3d where a is the first
term and d is the common difference.
The nth term of an arithmetic series is calculated as follows: a + d(n-1), where a is the first term
and d is the common difference. The common difference is multiplied by (n-1) rather than n
because you add the common difference the first time to create the second term, the second
time to create the third term and so on.
The sum of the first n terms in an arithmetic sequence can be calculated as follows: n(2a +
d(n-1))/2 where a is the first term, d is the common difference, and n is the number of terms.
This formula works by taking an average of the first term and the nth term and multiplying this
average by the number of terms. Because the series increases by regular intervals, the average
of the first term and the last term is equal to the average of all the terms, so the sum of the
sequence can be found by multiplying the average of the first and last terms by n, the number
of terms in the sequence. Think about this formula in terms of the formula for the nth term of
an arithmetic sequence and your knowledge of averages to fully understand why it works.
1) What is the 30th term of the following series: 8, 15, 22, 29…?
You can tell this series is arithmetic because the same number (7) is being added to every term.
Thus, 7 is the common difference. Plug the first term, 8, and the common difference, 7, into
the formula for the nth term of an arithmetic sequence: a + d(n-1). 8 + 7(30 – 1) 8 + 7(29) =
211.
2) What is the 12th term of the following series: 5, -1, -7, -13…?
You can tell this series is arithmetic because the same number (-6) is being added to every
term. Thus, -6 is the common difference. Plug the first term, 5, and the common difference, -6,
into the formula for the nth term of an arithmetic sequence: a + d(n-1). 5 + -6(12 – 1) 5 –
6(11) = -61.
3) What is the sum of the first 15 terms of the following sequence: 6, 10, 14, 18…?
You can tell that this sequence is arithmetic because the same number (4) is added to each
term to get the next term. Thus, 4 is the common difference of this sequence. Plug the first
term, 6, and the common difference, 4, into the formula for the sum of the first n terms of an
arithmetic sequence, n(2a + d(n-1))/2. 15(2 • 6 + 4(15 – 1))/2 = 15(12 + 4 • 14)/2 = 15(68)/2 =
510.
You could also do this problem conceptually. To do so, find the last term in the arithmetic
sequence using the formula for the nth term in an arithmetic sequence. The 15 th term is equal
to 6 + 4(15 – 1) = 6 + 56 = 62. Now take the average of the first term, 6, and the nth term, 62. (6
+ 62)/2 = 34. Because this series is arithmetic and therefore increases at a constant rate, the
average of the first and last terms is equal to the overall average of all the terms, so you can
multiply this average by n to find the sum of the sequence. 34 • 15 = 510.
4) A pizza shop sells 10 pizzas its first day in business. Each day after the first day, the pizza
shop sells 7 more pizzas than it did the day before. How many pizzas does it sell on the 18 th
day?
This question is asking you to find the 18th term of an arithmetic series with a first term of 10
and a common difference of 7. Plugging these numbers into the formula for the nth term of an
arithmetic sequence, you get 10 + 7(18 – 1) = 10 + 7(17) = 129. You could also conceive of this
problem conceptually, realizing that you add the first 7 on the second day, the second 7 on the
third day, and so on. Therefore, on the 18th day, you must add the 17th 7, so the answer is 10 +
7(17) = 129.
5) A pizza shop sells 10 pizzas its first day in business. Each day after the first day, the pizza
shop sells 7 more pizzas than it did the day before. How many pizzas does it sell in the first 18
days?
It is crucial to recognize the subtle difference in wording between this problem and problem
four, since this difference completely changes the problem. Problem 4 involved finding the n th
term of the sequence, while this problem involves finding the sum of the first n terms. To find
the sum of this sequence, plug the first term of 10, the common difference of 7, and the total of
18 terms into the formula for the sum of the first n terms of an arithmetic sequence. 18(2 • 10
+ 7(18-1))/2 = 18(20 + 119)/2 = 1251.
You could also do this problem conceptually. To do so, find the last term in the arithmetic
sequence using the formula for the nth term in an arithmetic sequence. As demonstrated in
problem 4, the 18th term is equal to 129. Now take the average of the first term, 10, and the
nth term, 129. (10 + 129)/2 = 69.5. Because this series is arithmetic and therefore increases at
a constant rate, the average of the first and last terms is equal to the overall average of all the
terms, so you can multiply this average by n to find the sum of the sequence. 69.5 • 18 = 1251.
Geometric Series
A geometric series is a series in which each term is multiplied or divided by the same number to
produce the next term. The number that each term in the series is multiplied by to produce the
next term is known as the common ratio. If you divide any term by the previous term, the
result is equal to the common ratio. For instance, 10, 20, 40, 80… is a geometric series with a
first term of 10 and a common ratio of 2. 405, 135, 45, 15… is a geometric series with a first
term of 405 and a common ratio of 1/3.
The terms of a geometric series progress as follows: a, ar, ar2, ar3 where a is the first term and
r is the common ratio. Note that only r (and not a) is being raised to a power.
The nth term of a geometric series is calculated as follows: ar(n-1), where a is the first term and
d is the common difference. The common difference is multiplied by (n-1) rather than n
because you multiply by the common difference the first time to create the second term, the
second time to create the third term, and so on.
The sum of the first n terms in a geometric sequence can be calculated as follows: a(1-rn)/(1r). This concept is not truly essential to learn, as it comes up almost never on the SAT or ACT.
The sum of an infinite geometric sequence can be calculated as follows: a/(1-r). This concept
is seen more frequently on the SAT and ACT than the partial sum described above. On the ACT,
you will typically be given this formula if you need to use it.
6) What is the 9th term of the following series: 3, 12, 48, 192…?
You can tell this series is geometric because each term is multiplied by the same number (4) to
produce the next term. Thus, 4 is the common ratio. Plug the first term, 3, and the common
ratio, 4, into the formula for the nth term of a geometric sequence: ar(n-1) 3 • 49-1 = 3 • 48 =
196608.
7) What is the 15th term of the following series: 1,-2, 4, -8…?
You can tell this series is geometric because each term is multiplied by the same number (-2) to
produce the next term. Thus, -2 is the common ratio. Plug the first term, 1, and the common
ratio, -2, into the formula for the nth term of a geometric sequence: ar (n-1) 1 • (-2)15-1 = 1 • (2)14 = 16384.
8) What is the sum of the first 8 terms of the following sequence: 2, 6, 18, 54…?
You can tell this sequence is geometric because each term multiplied by the same number (3)
to produce the next term. Thus, 3 is the common ratio. Plug the first term, 2, the common
ratio, 3, and the number of terms, 8, into the equation for the sum of the first n terms of a
geometric series, a(1-rn)/(1-r). 2(1-38)/(1-3) = 2(-6561)/-2 = 6561.
9) What is the sum of the following infinite geometric sequence: 28, 14, 7, 3.5…?
You can tell this sequence is geometric because each term multiplied by the same number (1/2)
to produce the next term. Thus, 1/2 is the common ratio. Plug the first term, 28, and the
common ratio, 1/2, into the equation for the sum of an infinite geometric sequence, a/(1-r).
28/(1 – ½) = 28/.5 = 56.
Repeating Number Patterns
Occasionally, both the SAT and ACT will test you on repeating number patterns. Technically,
these are not series or sequences, but they are worth mentioning in the discussion of series and
sequences. Observe the following example.
10) What is the 53rd digit to the right of the decimal place in the following decimal:
.387538753875…?
To figure out a problem involving repeating numbers, first figure out how many digits are in
each repetition of the pattern. The pattern in this problem repeats every 4 digits. Therefore,
any multiple of 4 digits to the right of the decimal must be equal to the last digit in the pattern,
which is 5. The next step is to find a multiple of 4 that is close to (preferably just under) 53. 52
is a multiple of 4, so the 52nd digit to the right of the decimal place must be a 5. Therefore, the
53rd digit to the right of the decimal place must be the digit in the pattern that comes right after
5, which in this case is 3.