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# Arithmetic sequences review

Review arithmetic sequences and solve various problems involving them.

## Parts and formulas of arithmetic sequences

In arithmetic sequences, the difference between consecutive terms is always the same. We call that difference the common difference.
For example, the common difference of the following sequence is plus, 2:
start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6
3, comma5, comma7, comma9, comma, point, point, point
Arithmetic sequence formulas give a, left parenthesis, n, right parenthesis, the n, start superscript, start text, t, h, end text, end superscript term of the sequence.
This is the explicit formula for the arithmetic sequence whose first term is start color #11accd, k, end color #11accd and common difference is start color #ed5fa6, d, end color #ed5fa6:
a, left parenthesis, n, right parenthesis, equals, start color #11accd, k, end color #11accd, plus, left parenthesis, n, minus, 1, right parenthesis, start color #ed5fa6, d, end color #ed5fa6
This is the recursive formula of that sequence:
$\begin{cases}a(1) = \blueD k \\\\ a(n) = a(n-1)+\maroonC d \end{cases}$

## Extending arithmetic sequences

Suppose we want to extend the sequence 3, comma, 8, comma, 13, comma, point, point, point We can see each term is start color #ed5fa6, plus, 5, end color #ed5fa6 from the previous term:
start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6
3, comma8, comma13, comma, point, point, point
So we simply add that difference to find that the next term is 18:
start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6
3, comma8, comma13, comma18, comma, point, point, point
Problem 1
• Current
What is the next term in the sequence minus, 5, comma, minus, 1, comma, 3, comma, 7, comma, dots?

Want to try more problems like this? Check out this exercise.

## Writing recursive formulas

Suppose we want to write a recursive formula for 3, comma, 8, comma, 13, comma, point, point, point We already know the common difference is start color #ed5fa6, plus, 5, end color #ed5fa6. We can also see that the first term is start color #11accd, 3, end color #11accd. Therefore, this is a recursive formula for the sequence:
$\begin{cases}a(1) = \blueD 3 \\\\ a(n) = a(n-1)\maroonC{+5} \end{cases}$
Problem 1
• Current
Find k and d in this recursive formula of the sequence minus, 5, comma, minus, 1, comma, 3, comma, 7, comma, dots.
$\begin{cases}a(1) = k \\\\ a(n) = a(n-1)+d \end{cases}$
k, equals
d, equals

Want to try more problems like this? Check out this exercise.

## Writing explicit formulas

Suppose we want to write an explicit formula for 3, comma, 8, comma, 13, comma, point, point, point We already know the common difference is start color #ed5fa6, plus, 5, end color #ed5fa6 and the first term is start color #11accd, 3, end color #11accd. Therefore, this is an explicit formula for the sequence:
a, left parenthesis, n, right parenthesis, equals, start color #11accd, 3, end color #11accd, start color #ed5fa6, plus, 5, end color #ed5fa6, left parenthesis, n, minus, 1, right parenthesis
Problem 1
• Current
Write an explicit formula for minus, 5, comma, minus, 1, comma, 3, comma, 7, comma, dots
a, left parenthesis, n, right parenthesis, equals

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• How do you write an explicit formula for the sequence (8,11,16,23,32)
• Notice that the sequence of differences is simply the sequence of odd numbers. So this is the sequence of squares shifted by a constant. Since 8-1=7, we have
a(n) = n^2 +7
• Why was the Arithmetic Sequence review so hard on my mind? It took me to look at the explanation to understand the answers to every darn question. And, I am yet so confused on this section.
• I watched all the videos and I'm ready for my semester exams. The concept is really easy once you get to know it.
• When given an arithmetic sequence with a negative term, how do you calculate what term the number will be when it reaches 0?
• Some sequences do not even have 0 as one of the possibilities. So if you start with a negative and have a negative common difference, then there is no zero such as -2, -4, -6, -8, ... Others will have a common difference that skips over zero such as -4, 4, 12, 20, 28, ... If the sequence does include a zero, such as an initial value of - 9 and a common difference of 3, we get an equation f(n) = -9 + 3(n-1). set this equal to zero, 0 = -9 + 3(n-1) move the - 9 by adding and distribute to get 9 = 3n - 3, add 3 to both sides, 3n = 12., divide by 3 to get n = 4. Sequence would be -9, -6, -3, 0 ... which is what we were looking for. So if your equation skipped 0 as above f(n) = -4 +8(n-1) and did the same thing, then 0 = -4 + 8n - 8, add 12 to get 12 = 8n, n = 12/8 = 3/2, but a sequence does not have a 3/2 term, only 1, 2, 3 ... terms.
• How would you find the explicit formula for a problem such as:
"a" subscript 38 =-53.2, while the common difference is -1.1?
• You need to know a common difference and one 'value' (sequence) in order to find the general form (formula). Since the common difference is -1.1, it would look like this: -1.1(n-1) + b (if you are starting from n=1).
Then plug in "a" subscript 38 = -53.2 into the formula. Then it would be -1.1(38-1) + b = -53.2
All you have to do is solving for 'b' and it would be -12.5.
Therefore, the general form (formula) would be -1.1(n-1) - 12.5 if you are starting from n=1.
• how are the formulas for arithmetic sequences similar to functions?
• Arithmetic sequences are functions. The difference between a linear function and a arithmetic sequence is that the first is continuous and the second is discrete, but the continuous line would go through all the points of its equivalent sequence.
• how do you write a recursive formula for 1, 1, 0, -1, -1, 0, 1, 1
• Consider the arithmetic sequence 27, 13, -1, ...
The explicit rule for the sequence in terms of n is a(n)= 27-14(n-1). If the nth term is -841, find the value of n.