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 +2+2:
+2\footnotesize\maroonC{+2\,\Large\curvearrowright}+2\footnotesize\maroonC{+2\,\Large\curvearrowright}+2\footnotesize\maroonC{+2\,\Large\curvearrowright}
3,3,5,5,7,7,9,...9,...
Arithmetic sequence formulas give a(n)a(n), the nthn^{\text{th}} term of the sequence.
This is the explicit formula for the arithmetic sequence whose first term is k\blueD k and common difference is d\maroonC d:
a(n)=k+(n1)da(n)=\blueD k+(n-1)\maroonC d
This is the recursive formula of that sequence:
{a(1)=ka(n)=a(n1)+d\begin{cases}a(1) = \blueD k \\\\ a(n) = a(n-1)+\maroonC d \end{cases}
Want to learn more about arithmetic sequences? Check out this video.

Extending arithmetic sequences

Suppose we want to extend the sequence 3,8,13,...3,8,13,... We can see each term is +5\maroonC{+5} from the previous term:
+5\maroonC{+5\,\Large\curvearrowright}+5\maroonC{+5\,\Large\curvearrowright}+5\maroonC{+5\,\Large\curvearrowright}
3,3,8,8,13,...13,...
So we simply add that difference to find that the next term is 1818:
+5\maroonC{+5\,\Large\curvearrowright}+5\maroonC{+5\,\Large\curvearrowright}+5\maroonC{+5\,\Large\curvearrowright}
3,3,8,8,13,13,18,...18,...
Problem 1
What is the next term in the sequence 5,1,3,7,-5,-1,3,7, \ldots?
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

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

Writing recursive formulas

Suppose we want to write a recursive formula for 3,8,13,...3,8,13,... We already know the common difference is +5\maroonC{+5}. We can also see that the first term is 3\blueD3. Therefore, this is a recursive formula for the sequence:
{a(1)=3a(n)=a(n1)+5\begin{cases}a(1) = \blueD 3 \\\\ a(n) = a(n-1)\maroonC{+5} \end{cases}
Problem 1
Find kk and dd in this recursive formula of the sequence 5,1,3,7,-5,-1,3,7, \ldots.
{a(1)=ka(n)=a(n1)+d\begin{cases}a(1) = k \\\\ a(n) = a(n-1)+d \end{cases}
k=k=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
d=d=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

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

Writing explicit formulas

Suppose we want to write an explicit formula for 3,8,13,...3,8,13,... We already know the common difference is +5\maroonC{+5} and the first term is 3\blueD3. Therefore, this is an explicit formula for the sequence:
a(n)=3+5(n1)a(n)=\blueD3\maroonC{+5}(n-1)
Problem 1
Write an explicit formula for 5,1,3,7,-5,-1,3,7, \ldots
a(n)=a(n)=

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